Definition of Exchange Rate Determinations and Forecasting

Introduction

The rupee/dollar rate is a two-way rate which means that the price of 1 dollar is quoted in terms of how much rupees it takes to buy one dollar. The value of one currency against another is based on the demand of the currency. If the demand for dollar increases, the value of dollar would appreciate. As the quotation for Rs/$ is a two way quote, an appreciation in the value of dollar would automatically mean the depreciation in Indian rupee and vice-versa. For example if rupee would depreciate, a dollar which once cost ` 47 would cost say ` 59. So in essence the value of dollar has risen and the buying power of the rupee has gone down. Besides the primary powers of demand and supply, the rupee-dollar rates are determined by other market forces as well such as:

Market Sentiments

During turbulent markets, investors usually prefer to park their money in safe havens such as US treasuries, Swiss Franc, gold in order to avoid losses to their portfolios.

So this flight to safety would lead to foreign investors redeeming their investments from India and would naturally increase the demand for dollar vis-à-vis the Indian rupees. Remember the rupee/dollar rates during 2007 and 2008? Even today we are seeing a lot of FIIs redeeming their investments from emerging markets like India and are investing into US treasuries which are currently quoting at higher yields. This has lead to Indian rupee depreciating to ` 60/$.

Speculation

When the markets are moving vertically, there’s a lot of speculation about the expected changes into the currency rates due to the investments/redemptions of foreign investors. There are derivative instruments and over-the-counter currency instruments through which one can speculate/hedge the underlying currency rates. When speculators can sense improvements/deterioration of the sentiments of the markets, they too want to benefit from such rising/falling dollar and they start buying/selling dollar which would further increase the demand/supply of dollar.

RBI Intervention

When there is too much volatility in the rupee-dollar rates, the RBI prevents rates going out of control to protect the domestic economy. The RBI does this by buying dollars when the rupee appreciates too much and by selling dollars when the rupee depreciates way too much. The same was recently felt on June 12, 2013 when the rupee recovered sharply from ` 58.95/$ level.

Imports and Exports

Ever thought why our Government is trying to incentivize exports and reduce imports?

There are a lot of schemes and incentives for exporters while importers are burdened by many conditions and taxes. This is to protect our economy from high rupee depreciation. Importing foreign goods requires us to make payment in dollars thus strengthening the dollar’s demand and exports do the reverse. Major imports being fuel and gold; understandably even today we are a net-importing country which means that we are importing more and exporting less.

Interest Rates

The interest rates on Government bonds in emerging countries such as India attract foreign capital to India.

If the rates are high enough to cover foreign market risk and if the foreign investor/ fund is comfortable with the Sovereign’s fundamentals/credit ratings, money would start pouring in India and thus would provide a fillip to rupee demand.

Short-Run Forecasting Tools

Short-term changes in exchange rates are the most difficult to predict and are often determined based on bandwagon effects, overreaction to news, speculation, and technical analysis.

Trend-Following Behavior is the tendency for the market to follow a trend. In other words an increase in the exchange rate is more likely to be followed by another increase.

Investor Sentiment is based on the consensus of the market. For example if the market is bullish on the dollar, then the dollar is likely to strengthen versus other currencies.

The FX market is quite different from the world equity markets in one important aspect: transparency. In equity markets, rules ensure that volume and price data are readily available to all parties… this is NOT the case in FX markets. In fact large FX dealers are able to observe factors such as: shifts in risk appetite, liquidity needs, hedging demands, and institutional rebalancing.

Order Flow

There is evidence of a positive correlation between spot exchange rate movements and order flows in the inter-dealer market and with movements in customer order flows.

Three explanations for the cause of these correlations have been put forth:

Private information - related to the payoff from holding the currency may be contained in the order flow data. For example, future interest rates or the discount rate may be known to traders.

Liquidity effects – dealers charge a temporary risk premium to absorb unwanted inventory.

Feedback trading – the positive correlation could be related to customers buying a currency that has just appreciated (or vice versa).

Long-Run Forecasting Tools

Purchasing Power Parity (PPP) states that since the prices should be the same across countries, the exchange rate between two countries should be the ratio of the prices in each country.

Relative PPP states that the exchange rate will change to offset differences in national interest rates. In other words, if Country A has higher inflation than Country B, you can expect Country 

A’s currency to depreciate versus Country B’s currency.

Structural Changes

Three structural changes can affect long-term trends in exchange rates: 1) an increase in investment spending, 2) fiscal stimulus, 3) a decline in private savings.

It is the net impact of structural changes that determines if the country’s currency will rise or fall.

Investment spending – domestic investment in a country will help to strengthen a country’s currency. For example, the United States experienced an investment boom in the 1990s.

Fiscal stimulus – government investment in a country can also help strengthen a country’s currency. For example, Turkey has enjoyed fiscal stimulus and government spending in recent years.

Private savings – the citizens of a country’s tendency to save will help strengthen a country’s currency. For example, Japan has had a large and persistent current-account surplus that has led to a stronger currency.

Terms of Trade

Is the idea that the price of a good that trades in international markets will have an impact of the associated country’s currency. This can work in terms of both imports and exports.

For example, in countries where commodities make up a large portion of GDP, like Australia, Canada, and New Zealand, there is a strong positive relationship between the price of commodities and the strength of the associated country’s currency. On the other hand, in Europe, the higher prices for oil, have led to a weaker currency.

Medium-Run Forecasting Tools

International Parity Conditions

The key international parity conditions are 1) purchasing power parity, 2) covered interest-rate parity, 3) uncovered interest-rate parity, 4) the Fisher effect, and 5) forward exchange rates. 

Purchasing power parity – states that since the prices should be the same across countries, the exchange rate between two countries should be the ratio of the prices in each country.

Example: If a hamburger is $2.54 in the United States and 3.60 real (R$) in Brazil, then the PPP spot rate should be:

FYI McDonalds’ Big Mac is produced locally in almost 120 countries!

Covered interest-rate parity –the idea that an imbalance in parity conditions can create a “risk less” opportunity for an arbitrager.

                                                                                                           Covered Interest Arbitrage (CIA)

Example

Step 1: Convert $1,000,000 at the spot rate of ¥106.00/$ to ¥106,000,000

Step 2: Invest the proceeds, (¥106,000,000), in a euroyen account for six months, earning 4% per annum, or 2% for 180 days.

Step 3: Simultaneously sell the future yen proceeds (¥108,120,000) forward for dollars at the 180-day forward rate of ¥103.50/$. Note: at this point you have “locked in” the amount of $1,044,638 in 180 days (or 6 months).

Step 4: Out of the $1,044,638 you have to repay the loan (plus interest), this is called your opportunity cost of capital. To do this, calculate the interest rate for the period (8% per year is 4% for 180 days). So to borrow $1,000,000 you have to pay $40,000 in interest at the end of 6 months. Subtract the $1,040,000 from the $1,044,638 that you will receive from your forward contract for a “risk less” profit of $4,638.

Notice that these activities should help the currencies return to equilibrium.

Since there are men and women making a killing in this business, the opportunities for smaller investors are almost impossible… It is these two types of arbitrage that keep exchange rates more or less in equilibrium.

Fisher effect - the nominal interest rate (i) in a country should be equal to the real rate of interest (r) plus expected inflation (π).

     i = r + π

Forward exchange rates – an exchange rate quoted today for settlement at a future date.

Forward rates are unbiased predictors of future exchange rates. An unbiased predictor means that “on average” the estimation will be wrong on the up side or the downside with equal frequency and degree. In other words, the errors are normally distributed.

Forecasting Exchange Rates

One of the goals of studying the behavior of exchange rates is to be able to forecast exchange rates. Chapters III and IV introduced the main theories used to explain the movement of exchange rates. These theories fail to provide a good approximation to the behavior of exchange rates. Forecasting exchange rates, therefore, seems to be a difficult task.

This chapter analyzes and evaluates the different methods used to forecast exchange rates. This chapter closes with a discussion of exchange rate volatility.

Forecasting Exchange Rates

International transactions are usually settled in the near future. Exchange rate forecasts are necessary to evaluate the foreign denominated cash flows involved in international transactions. Thus, exchange rate forecasting is very important to evaluate the benefits and risks attached to the international business environment.

A forecast represents an expectation about a future value or values of a variable. The expectation is constructed using an information set selected by the forecaster. Based on the information set used by the forecaster, there are two pure approaches to forecasting foreign exchange rates:

The fundamental approach.

The technical approach.

Fundamental Approach

The fundamental approach is based on a wide range of data regarded as fundamental economic variables that determine exchange rates. These fundamental economic variables are taken from economic models. Usually included variables are GNP, consumption, trade balance, inflation rates, interest rates, unemployment, productivity indexes, etc. In general, the fundamental forecast is based on structural (equilibrium) models. These structural models are then modified to take into account statistical characteristics of the data and the experience of the forecasters. It is a mixture of art and science.

Practitioners use structural model to generate equilibrium exchange rates. The equilibrium exchange rates can be used for projections or to generate trading signals. A trading signal can be generated every time there is a significant difference between the model-based expected or forecasted exchange rate and the exchange rate observed in the market. If there is a significant difference between the expected foreign exchange rate and the actual rate, the practitioner should decide if the difference is due to a mispricing or a heightened risk premium. If the practitioner.

Fundamental Approach: Forecasting at Work

The fundamental approach starts with a model, which produces a forecasting equation. This model can be based on theory, say PPP, a combination of theories or on the ad-hoc experience of a practitioner. Based on this first step, a forecaster collects data to estimate the forecasting equation. The estimated forecasting equation will be evaluated using different statistics or measures. If the forecaster is happy with the model, she will move to the next step, the generation of forecasts. The final step is the evaluation of the forecast.

As mentioned above, a forecast represents an expectation about a future value or values of a variable. In this chapter, we will forecast a future value of the exchange rate, St+T. The expectation is constructed using an information set selected by the forecaster. The information set should be available at time t. The notation used for forecasts of St+T is:

Et [St+T],

where Et[.] represent an expectation taken at time t.

Each forecast has an associated forecasting error, εt+1. We will define the forecasting error as:

εt+1= St+1 - Et[St+1]

The forecasting error will be used to judge the quality of the forecasts. A typical metric used for this purpose is the Mean Square Error or MSE. The MSE is defined as:

MSE = [(εt+1)2 + (εt+2)2 + (εt+3)2 + ... + (εt+Q)2]/Q,

Where Q is the number of forecasts, we will say that the higher the MSE, the less accurate the forecasting model. There are two kinds of forecasts: in-sample and out-of-sample. The first type of forecasts works within the sample at hand, while the latter works outside the sample. In-sample forecasting does not attempt to forecast the future path of one or several economic variables.

In-sample forecasting uses today›s information to forecast what today›s spot rates should be. That is, we generate a forecast within the sample (in-sample). The fitted values estimated in a regression are in-sample forecasts. The corresponding forecast errors are called residuals or in-sample forecasting errors.

On the other hand, out-of-sample forecasting attempts to use today are information to forecast the future behavior of exchange rates. That is, we forecast the path of exchange rates outside of our sample. In general, at time t, it is very unlikely that we know the inflation rate for time t+1. That is, in order to generate out-of-sample forecasts, it will be necessary to make some assumptions about the future behavior of the fundamental variables.

Summary: Fundamental Forecasting Steps

Selection of Model (for example, PPP model) used to generate the forecasts.

Collection of S_t, X_t (in the case of PPP, exchange rates and CPI data needed.)

Estimation of model, if needed (regression, other methods)

Generation of forecasts based on estimated model. Assumptions about X_t+T may be needed.

Evaluation. Forecasts are evaluated. If forecasts are very bad, model must be changed.

Example

In-sample Forecasting Exchange Rates with PPP

Suppose you work for a U.S. firm. You are given the following quarterly CPI series in the U.S. and in the U.K. from 2008:1 to 2009:3. The exchange rate in 2008:1 is equal to 1.9754 USD/GBP. You believe that this exchange rate, 1.5262 USD/GBP, is an equilibrium rate. Your job is to generate equilibrium exchange rates using PPP. In order to do this, you do quarterly in-sample forecasts of the USD/GBP exchange rate using relative PPP. That is,

Some calculations for S^F_2008:2 and S^F_2008:3:

Forecast S^F_2008:2.

I_US,2008:2  =  (USCPI_2008:2/USCPI_2008:1)  -  1  =  (111.0/108.6)  -  1  =  0.0221.  I_UK,2008:2

= (UKCPI_2008:2/UKCPI_2008:1) - 1 = (108.2/106.2) - 1 = 0.0191. s^F_2008:2 = I_US,2008:2 - I_UK,2008:2

= 0.0221 - 0.0191 = 0.0030.

S^F_2008:2 = S^F_2008:1 x [1 + s^F_2008:2] = 1.9754 USD/GBP x [1 + (0.0030)] = 1.9813 USD/GBP.

ε_2008:2 = S^F_2008:2-S_2008:2 = 1.9813 – 1.9914 = -0.01.

Forecast S^F_2008:3.

S^F_2008:3 = S^F_2008:2 x [1 + s^F_2008:3] = 1.9914 USD/GBP x [1 + (0.0019)] = 1.9951 USD/GBP.

ε_2008:3 = S^F_2008:3-S_2008:3 = 1.9951 – 1.7705 = 0.2246.

Evaluation of forecasts.

MSE: [(-0.01)2 + (0.2246)2 + (0.2964)2 + .... + (0.0463)2]/6 = 0.0306

Now, you can generate trading signals. According to this PPP model, the equilibrium exchange rate in 2008:2 should be 1.9813 USD/GBP.

The market price, however, is 1.9914 USD/GBP. That is, the market is valuing the GBP higher than your fundamental model. Suppose you believe that the difference (1.9813-1.9914) is due to miss-pricing factors, then you will generate a sell GBP signal.

In general, practitioners will divide the sample in two parts: a longer sample (estimation period) and a shorter sample (validation period). The estimation period is used to select the model and to estimate its parameters.

Suppose we are interested in one-step-ahead forecasts. The one-step-ahead forecasts made in this period are in-sample forecasts, not “true forecasts.” These one-step-ahead forecasts are just fitted values. The corresponding forecast errors are called residuals.

The data in the validation period are not used during model and parameter estimation. One-step-ahead forecasts made in this period are “true forecasts,” often called backtests. These true forecasts and their error statistics are representative of errors that will be made in forecasting the future.

A forecaster will use the results from this validation step to decide if the selected model can be used to generate outside the sample forecasts.

Example: 2

Out-of-sample Forecasting Exchange Rates with PPP

Go back to Example V.1. Now, you want to generate out-of-sample forecasts.

You need to make some assumptions about the future behavior of the inflation rate.

Naive assumption: E_t[I_t+1] = I^F_t+1 = I_t.

You can generate out-of-sample forecasting by assuming that today’s inflation is the best predictor for tomorrow’s inflation. That is, E_t[I_t+1] = I^F_t+1 = I_t.

This “naive” forecasting model leads us to a simplified version of the Relative PPP:

E_t[s_t+1] = s^F _t+1 =(E_t[S_t+1]/S_t) – 1 ≈ I_d,t - I_f,t.

With the above information we can predict S_2008:3:

s^F_2008:3 = I_US,2008:2 - I_UK,2008:2 = 0.0221 - 0.0191 = 0.0030.

S^F_2008:3 = S_2008:2 x [1 + s^F_2008:3] = 1.9914 x [1 + (.0030)] = 1.99735

Autoregressive model: E[I_t+1] = α₀ + α₁ I_t.

More sophisticated out-of-sample forecasts can be achieved by estimating regression models, using survey data on expectations of inflation, etc. For example, consider the following regression model:

I_US,t = α^US₀ + α^US₁ I_US,t-1 + ε_US.t.

I_UK,t = α^UK₀ + α^UK₁ I_UK,t-1 + ε_UK,t.

This autoregressive model can be estimated using historical data, say 1978:1-2008:1.

Then, we have 119 quarterly inflation rates for both series. We estimate both equations.

Excel output for autoregressive model for the US.

Excel output for autoregressive model for the UK.

First, you evaluate the regression by looking at the t-statistics and the R2. The t-statistic is used to test the null hypothesis that a coefficient is equal to zero. The R2 measures how much of the variability of the dependent variables is explained by the variability of the independent variables. That is, the R2 measures the explanatory power of our regression model. Both R2 coefficients are far from zero, relatively high for the U.S. inflation rate (51%). All coefficients have a t-stats higher than 1.96. That is, you will say that they are significant at the 5% level –i.e., with p-values smaller than .05. That is, all the coefficients are statistically different from zero.

Second, you use the regression to forecast inflation rates. Then, you will use these inflation rate forecasts to forecast the exchange rate. That is,

I^F_US,2008:3 = .00292 + .7001 x (.0221) = .01839

I^F_UK,2008:3 = .00713 + .4144 x (.0191) = .01505

s^F_2008:3 = I^F_US,2008:3 - I^F_UK,2008:3 = .01839 - 01505 = .00334.

S^F_2008:3 = 1.9914 USD/GBP x [1 + (.00334)] = 1.99802 USD/GBP.

That is, you predict, over the next quarter, an appreciation of the GBP. You can use this information to manage currency risk at your firm.

For example, if, during the next quarter, the U.S. firm you work for expects to have GBP outflows, you can advise management to hedge.

Example: 3

Out-of-sample Forecasting Exchange Rates with a Structural Ad-hoc Model

Suppose a Malaysian firm is interested in forecasting the MYR/USD exchange rate. This Malaysian firm is an importer of U.S. goods. A consultant believes that monthly changes in the MYR/USD exchange rate are driven by the following econometric model (MYR = Malaysian Ringitt)

s_MYR/USD,t = a₀ + a₁ INF_t + a₂ INC_t + ε_t,        (V.1)

Where, INF_t represents the inflation rate differential between Malaysia and the U.S., and INC_t represents the income growth rates differential between Malaysia and the U.S.

The spot rate this month is S_t=3.1021 MYR/USD. Suppose equation (V.1) is estimated using 10 years of monthly data with ordinary least squares (OLS). We have the following excel output:

That is, the coefficient estimates are: a₀ = 0.00693, a₁ = 0.21593, and a₂ = 0.09159.

That is, the output from your OLS regression is:

The goal of a MA model is to smooth erratic daily swings of asset prices Let’s evaluate our ad-hoc model. The t-statistics (in parenthesis) for the two variables are all bigger than 1.65. Therefore, all the explanatory variables are statistically significant at the 10% level. This regression has an R2 equal to .0186. That is, INF and INC explain less than two percent of the variability of changes in the MYR/USD exchange rate. This is not very high, but the t-stats give some hope for the model. The t-statistics (in parenthesis) for the two variables are all bigger than 1.65. Therefore, all the explanatory variables are statistically significant at the 10% level. The Malaysian firm decides to use this model to generate out-of-sample forecasts.

Suppose the Malaysian firm has forecasts for next month for INF_t and INC_t: 3% and 2%, respectively. Then,

s^F_{MYR/USD,t+one-month} = 0.0069 + 0.21593 x (0.03) + .09159 x (0.02) = .0152.

The MYR is predicted to depreciate 1.52% against the USD next month. The spot rate this month is S_t=3.1021 MYR/USD, then, for next month, we predict:

S^F_t+1 = 3.1021 MYR/USD (1.0152) = 3.1493 MYR/USD.

Based on these results, the Malaysian firm, which imports goods from the U.S., decides to hedge its next month USD anticipated outflows.

Some Practical Issues in Fundamental Forecasting

There are several practical issues associated with any fundamental analysis forecasting, such as the forecasting model of equation (V.1):

Correct specification. That is, are we using the “right model?” (In econometrics jargon, “correct specification.”)

Estimation of the model. This is not a trivial issue. For example, in equation (V.1) we need to estimate the model to get a₀, a₁, and a₂. Bad estimates of a₀, a₁, and a₂ will produce a bad forecast for s_{MYR/USD,t+one-month}. This issue sometimes is related to (1).

Contemporaneous variables. In a model like equation (V.1), some of the explanatory variables are contemporaneous. We also need a model to forecast the contemporaneous variables. For example, in the equation (V.1) we need a model to forecast INT_t and INC_t. In econometrics jargon, this is called simultaneous equations models.

Technical Approach

The technical approach (TA) focuses on a smaller subset of the available data. In general, it is based on price information.

The analysis is “technical” in the sense that it does not rely on a fundamental analysis of the underlying economic determinants of exchange rates or asset prices, but only on extrapolations of past price trends. Technical analysis looks for the repetition of specific price patterns. Technical analysis is an art, not a science.

Computer models attempt to detect both major trends and critical, or turning, points. These turning points are used to generate trading signals: buy or sell signals.

The most popular TA models are simple and rely on moving averages (MA), filters, or momentum indicators.

Technical Analysis Models

MA Models

in order to signal major trends. A MA is simply an average of past prices. We will use the simple moving average (SMA).

An SMA is the unweighted mean of the previous Q data points: SMA = (S_t + S_t-1 + S_t-2 + ... + S_t-(Q-1))/Q

If we include the most recent past prices, then we calculate a short-run MA (SRMA).

If we include a longer series of past prices, then we calculate a long-term MA (LRMA).

The double MA system uses two moving averages: a LRMA and a SRMA. A LRMA will always lag a SRMA because it gives a smaller weight to recent movements of exchange rates.

In MA models, buy and sell signals are usually triggered when a SRMA of past rates crosses a LRMA. For example, if a currency is moving downward, its SRMA will be below its LRMA. When it starts rising again, it soon crosses its LRMA, generating a buy foreign currency signal.

Buy FC signal: When SRMA crosses LRMA from below.

Sell FC signal: When SRMA crosses LRMA from above.

Example V.5

Generating trading signals for the (USD/GBP) using the Double MA model. We generate a SRMA using 30 days of information (red line)

We generate a LRMA using 150 days of information (green line).

Every time there is a crossing, the double MA model generates a trading signal.

The double MA model generates many trading signals, as indicated by the crossings between the SRMA (red line) and the LRMA (green line). For example, there is a sell GBP signal in late 2007. By April 2009, the model generates a buy GBP signal.

Filter Models

This is probably the most popular TA model. It is based on the finding that asset prices show significant small autocorrelations. If price increases tend to be followed by increases and price decreases tend to be followed by decreases, trading signals can be used to profit from this autocorrelation. The key of the system relies on determining when exchange rates start to show significant changes, as opposed to irrelevant noisy changes. Filter methods generate buy signals when an exchange rate rises X percent (the filter) above its most recent trough, and sell signals when it falls X percent below the previous peak. Again, the idea is to smooth (filter) daily fluctuations in order to detect lasting trends. The filter size, X, is typically between 0.5% and 2.0%.

Example V.6

Determination of Trading signals with a filter model.

Let the filter, X, be 1% => X= 1%.

First, we need to determine a peak or a through. Then, we generate trading signals.

Peak = 1.486 CHF/USD (X = CHF .01486) → When S_t crosses 1.47114 CHF/USD, Sell USD

Trough = 1.349 CHF/USD (X = CHF .01349) → When S_t crosses 1.36249 CHF/USD, Buy USD.

Note that there is a trade-off between the size of the filter and transaction costs. Low filter values, say 0.5%, generate more trades than a large filter, say 2%. Thus, low filters are more expensive than large filters. Large filters, however, can miss the beginning of trends and then be less profitable.

Momentum Models

Momentum models determine the strength of an asset by examining the change in velocity of the movements of asset prices. If an asset price climbs at increasing speed, a buy signal is issued.

These models monitor the derivative (slope) of a time series graph. Signals are generated when the slope varies significantly. There is a great deal of discretionary judgement in these models. Signals are sensitive to alterations in the filters used, the period length used to compute MA models and the method used to compute rates of change in momentum.

Basic Forecasting Models

Forecasting from Econometric Models

The econometric approach to forecasting consists first of formulating an econometric model that relates a dependent variable to a number of independent variables that are expected to affect it. The model is then estimated and used to obtain conditional or unconditional forecasts of the dependent variable. The models are generally formulated using economic theory and the statistical properties of the variables included in the model.

Example A.V.1

In Example V.3, a company believes that monthly changes in the MYR/USD exchange rate are related to the interest rates differential between Malaysia and the U.S. (INT_t) and income growth rates differential (INC_t) between Malaysia and the U.S. That is, the econometric model is given by:

^sMYR/USD,t,one-month ^{= a}0 ⁺ δ ^INTt ⁺ µ ^INCt ⁺ εt^,    (A.1)

Where ε_t is a prediction error assumed to follow a normal distribution with zero mean and constant variance, σ2.

The IFE predicts that INT_t should have a positive coefficient. That is, if Malayan interst rates increase relative to U.S. interest rates, then the MYR should depreciate with respect to the USD (i.e., δ should be positive). Similarly, the Asset Approach predicts that INC_t should have a negative coefficient. That is, if income grows in Malaysia at a faster rate than in the U.S., the MYR should appreciate with respect to the USD (i.e., µ should be negative).

Several economic series seem to show seasonal effects. For example, many researchers have found a Monday effect in the U.S. stock market. Since these seasonal effects are predictable, many forecasters include seasonal variables in an econometric model like equation (A.1).

Example A.V.2

In Example A.V.1 a forecaster might like to introduce monthly seasonal variables to predict the monthly change in the MYR/USD. In this case, equation (A.1) would include eleven monthly dummy variables.

sMYR/USD,t,one-month = a0 + δ INTt + µ INCt + τJan DJan + ... + τNov DNov +εt,

where

Forecasting from Time Series Models

Econometric models are generally based on some underlying economic model. A popular alternative to econometric models, especially for short-run forecasting is known as time series models. These models typically relate a dependent variable to its past and to random errors that may be serially correlated.

Time series models are generally not based on any underlying economic behavior.

A powerful time series model is the ARMA (Autoregressive Moving Average) process. The basic idea is that the series s_t at time t is affected by past values of s_t in a predictable manner. A general ARMA(p,q) can be written as:

s_t = α₀ + α₁ s_t-1 + ... + α_p s_t-q + ß₁ ε_t-1 + ... + ß_p ε_t-q + ε_t,        (A.2)

where ε_t is the prediction error at time t assumed to have a constant variance σ2. The terms with the α›s coefficients are the moving average terms. The terms with the ß›s coefficients are the moving average terms.

In order for the ARMA model in (A.2) to have nice properties -i.e., to be stationary-, we need to check that the roots of the polynomial

1 - (α₁ z + α₂ z² + ... + α_p z^p) = 0

lie outside the unit circle. In general, this requires that |α_I| < 1.

The prediction error, ε_t, is just the difference between the realization of s_t and the prediction of s_t using the ARMA(p,q) model.

Example A.V.3

Suppose we estimate equation (A.2) and we obtain

s^p_t = a₀ + a₁ s_t-1 + ... + a_p s_t-q + b₁ ε_t-1 + ... + b_p ε_t-q,

Where s^p_t is the predicted change in s_t, the a_i’s are the estimated α_i’s coefficients, and the b_i’s are the estimated ß_i’s coefficients. Then, ε_t = s_t - s^p_t.

Note: Suppose that s_t represents changes in the MYR/USD exchange rate. According to (A.2), the past p changes in the MYR/USD exchange rate affect today’s change in the MRY/USD exchange rate. Also, the past q prediction errors affect today’s change in the MYR/USD exchange rate.

The key component of the ARMA model is to determine q and p. Several statistical packages provide identification tools to determine q and p.

Many forecasters prefer to work with simpler AR(p) models. In this case, to determine p, a simple rule of thumb can be followed: start with an AR(1) model and add terms until the added terms are not statistically significant.

Forecasting Using a Combination of Methods

Many forecasters use a combination of the methods described in A.I and A.II. The dependent variable might depend on theoretical grounds on a set of independent variables. On empirical grounds it has been found that the dependent variable shows a high degree of autocorrelation. Although this autocorrelation is not present in the economic model, an economist might combine an economic model with an ARMA model to produce a better forecast.

Example

Suppose a forecaster believes that changes in the monthly MYR/USD exchange rate are determined by the IFE. She also has found that an ARMA (1,1) helps to predict future changes in exchange rates. She decides to use the following forecasting model:

sMYR/USD,t,one-month = α0 + δ INTt + α1 st-1 + ß1 εt-1 + εt,

where ε_t is a prediction error with a constant variance, σ2.

A Stationarity and Trends in Macroeconomic and Financial Data

In the previous sections, we have implicitly assumed that the dependent variable and independent variables are stationary. Roughly speaking, stationarity implies that the unconditional moments of a time series are independent of time. That is, they are constant.

Example A.V.5

The process for Y_t is said to be weakly stationary if:

The assumption of stationarity might not be appropriate for many of the economic and financial series used in practice. Several economic and financial series show clear trends: GDP, Consumption, CPI prices, stock prices, exchange rates, etc. For example, in Figure V.2, the CHF/USD shows a clear, predictable positive trend. This trend should be incorporated into any forecasting model.

There are two ways to achieve stationarity for these non-stationary series. The idea is to incorporate this trend in the model: (1) a deterministic time trend and (2) stochastic trend. The first model, also referred as trend-stationary, includes a deterministic time trend. The second model, also referred as a unit root process, uses first differences instead of levels.

Example

Suppose y_t is a non-stationary series.

Trend-stationary process. y_t = α + δ t + ε_t,

where ε_t is a stationary error.

Unit Root process. y_t - y_t-1 = α + ε_t,

where ε_t is a stationary error. This simple process is called a random walk with drift α.

We should note that this unit root process can be written in an AR(1) form:

y_t = α₁ y_t-1 + α + ε_t,

where α₁=1.

Both processes have different implications. If the series y_t follows a trend-stationary process, a shock has a temporary effect on the series, and the series eventually catches up with its trend. On the other hand, if y_t follows a unit root process, a shock might have permanent consequences for the level of future y_t’s.

There are several tests to check if a series has a unit root. These tests usually find a unit root on all major macroeconomic data. Of particular interest to us, exchange rates, GNP, money supply, and price levels have unit roots. Therefore, it is highly advisable to estimate models for these series in first differences.

It is common to take logs of the data before using it (see the Appendix of the Review Chapter). For small changes, the first difference of the log of a variable is approximately the same as the percentage change in the variable:

log(y_t) - log(y_t-1) = log(y_t/y_t-1) = log[1 + (y_t-y_t-1)/y_t] ≈ (y_t-y_t-1)/y_t,

Where we have used the fact that for z close to zero, log(1+z) ≈ z. It is usually convenient to multiply log(y_t) by 100. Thus, the changes are measured in units of percentage change.

We should notice, however, that several economists claim that unit root tests are not very revealing. These economists claim that in finite samples -like the ones available to us-it is very difficult to distinguish between models with a unit root -i.e., α₁=1- and stationary models with α₁ very close to 1.

Interesting Readings

Appendix V is based on Introductory Econometrics with Applications, by Ramu Ramanathan, published by Harcourt Brace Jovanovich.

Appendix V-B: Taylor Rules

According to the Taylor rule, the CB raises the target for the short-term interest rate, i_t, if:

Inflation, I_t, raises above its desired level

Output, y_t, is above “potential” output

The target level of inflation is positive (deflation is thought to be worse than positive inflation for the economy). The target level of the output deviation is 0, since output cannot permanently exceed “potential output.”

John Taylor (1993) assumed the following reaction function by the CB:

where y-gap_t is the output gap –a percent deviation of actual real GDP from an estimate of its potential level-, and r* is the equilibrium level or the real interest rate, which Taylor assumes equal to 2%. The coefficients φ and γ are weights, which can be estimated (though, Taylor assumes them equal to .5).

Let I_t* and r* in equation BC.1 be combined into one constant term, µ = r* - φ I_t*. Then,

i_t = µ + λ I_t + γ y-gap_t,

where λ = 1 + φ.

For many countries, whose CB monitors S_t closely; the Taylor rule is expanded to include the real exchange rate, Rt:

i_t = µ + λ I_t + γ y-gap_t + δ R_t

Estimating this equation for the US and a foreign country can give us a forecast for the interest rate differential, which can be used to forecast exchange rates.

Exercises

Go back to Example V.2.

Take the autoregressive forecasting model, estimated above. What is S^F_1997:4?

Calculate the same forecast using the “naive” model.

Compare both forecasts with the in-sample PPP forecast.

You work for a Tunisian investment bank. You have available the following quarterly interest rate series in the U.S., i_USD, and in Tunisia, i_TND, from 1998:4 to 1999:3 (TND=Tunisian Dinar). The TND/USD in 1998:4 is equal to 1.1646. Your job is to do quarterly out-of-sample forecasts of TND/USD exchange rate for the period 1999:2 1999:3, using the linear approximation to the International Fisher Effect (IFE).

Generate one-step-ahead forecasts –that is, as new information arrives, a new next period forecast is generated- for the period 1999:1-1999:4.

Your firm uses the following forecasting regression model to forecast interest rates. Use a regression analysis.

i_USD,t = .0075 + .93 i_USD,t-1 +ε_t. i_TND,t = .0060 + .97 i_TND,t-1+ε_t.

Generate out-of-sample forecasts for the period 1999:1-1999:4.

Given that firms cannot forecast exchange rates, should they worry about currency risk?

J. Cruyff, a Dutch designer company, wants estimate the monthly volatility of the weekly EUR/USD exchange rate. They use the following AR (1)-GARCH(1,1) model:

This GARCH model is an asymmetric model. Negative shocks increase the variance more than positive shocks. The persistence parameter should be redefined. That is, λ=[α₁+ß₁+(1/2)δ].

Using data from January 1974 till August 1997, the «quants» at J. Cruyff estimated the model for s_t:

Find λ and calculate the unconditional variance, σ2. Is it well defined?

Given that e_{Aug 97} = -1.073, and σ²_{Aug 97} = 7.436, forecast the variance for September 1997.

Forecast the variance for August 1998.

You want to calculate the VAR of a position in EUR. The value of your position is USD 50 million. You estimated the volatility of changes in the USD/EUR exchange rates as 22%. The time interval is seven days. You use a 99% confidence interval to calculate.

Law of One Price

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The law of one price (LOP) is an economic concept which posits that “a good must sell for the same price in all locations” The law of one price constitutes the basis of the theory of purchasing power parity and is derived from the no arbitrage assumption.

Intuition

The intuition behind the law of one price is based on the assumption that differences between prices are eliminated by market participants taking advantage of arbitrage opportunities: Assume different prices for a single identical good in two locations, no transport costs and no economic barriers between both locations. The arbitrage mechanism can now be performed by both the supply and/or the demand site:

All sellers have an incentive to sell their goods in the higher-priced location, driving up supply in that location and reducing supply in the lower-priced location. If demand remains constant, the higher supply will force prices to decrease in the higher-priced location, while the lowered supply in the alternative location will drive up prices there.

Conversely, if all consumers move to the lower-priced location in order to buy the good at the lower price, demand will increase in the lower-priced location and - assuming constant supply in both locations - prices will increase, whereas the decreased demand in the higher-priced location leads the prices to decrease there. Both scenarios result in a single, equal price per homogeneous good in all locations. In efficient markets the convergence on one price is instant. (For further discussion, please refer to rational pricing).

Example: Financial Markets

Commodities can be traded on financial markets, where there will be a single offer price (asking price), and bid price. Although there is a small spread between these two values the law of one price applies (to each). No trader will sell the commodity at a lower price than the market maker’s bid-level or buy at a higher price than the market maker’s offer-level. In either case moving away from the prevailing price would either leave no takers, or be charity.

In the derivatives market the law applies to financial instruments which appear different, but which resolve to the same set of cash flows; see rational pricing. Thus:

“A security must have a single price, no matter how that security is created. For example, if an option can be created using two different sets of underlying securities, then the total price for each would be the same or else an arbitrage opportunity would exist.” A similar argument can be used by considering arrow securities as alluded to by Arrow and Debreu (1944).

Where the law does not apply

See also: Purchasing power parity Difficulties

The law does not apply inter temporally, so prices for the same item can be different at different times in one market. The application of the law to financial markets in the example above is obscured by the fact that the market maker’s prices are continually moving in liquid markets. However, at the moment each trade is executed, the law is in force (it would normally be against exchange rules to break it).

The law also need not apply if buyers have less than perfect information about where to find the lowest price. In this case, sellers face a tradeoff between the frequency and the profitability of their sales. That is, firms may be indifferent between posting a high price (thus selling infrequently, because most consumers will search for a lower one) and a low price (at which they will sell more often, but earn less profit per sale).

The Balassa-Samuelson effect argues that the law of one price is not applicable to all goods internationally, because some goods are not tradable. It argues that the consumption may be cheaper in some countries than others, because non tradable (especially land and labor) are cheaper in less developed countries. This can make a typical consumption basket cheaper in a less developed country, even if some goods in that basket have their prices equalized by international trade.

Apparent Violations

A well-known example of an apparent violation of the law was Royal Dutch/ Shell shares. After merging in 1907, holders of Royal Dutch Petroleum (traded in Amsterdam) and Shell Transport shares (traded in London) were entitled to 60% and 40% respectively of all future profits. Royal Dutch shares should therefore automatically have been priced at 50% more than Shell shares. However, they diverged from this by up to 15%.[4] This discrepancy disappeared with their final merger in 2005.

Purchasing Power Parity

GDP per capita by countries in 2013, calculated using PPP exchange rates.

In economics, purchasing power parity (PPP) is a component of some economic theories and is a technique used to determine the relative value of different currencies.

Theories that invoke purchasing power parity assume that in some circumstances (for example, as a long-run tendency) it would cost exactly the same number of, say, US dollars to buy euros and then to use the proceeds to buy a market basket of goods as it would cost to use those dollars directly in purchasing the market basket of goods.

The concept of purchasing power parity allows one to estimate what the exchange rate between two currencies would have to be in order for the exchange to be on par with the purchasing power of the two countries’ currencies.

Using that PPP rate for hypothetical currency conversions, a given amount of one currency thus has the same purchasing power whether used directly to purchase a market basket of goods or used to convert at the PPP rate to the other currency and then purchase the market basket using that currency. Observed deviations of the exchange rate from purchasing power parity are measured by deviations of the real exchange rate from its PPP value of 1.

ppp exchange rates help to avoid misleading international comparisons that can arise with the use of market exchange rates. For example, suppose that two countries produce the same physical amounts of goods as each other in each of two different years.

Since market exchange rates fluctuate substantially, when the GDP of one country measured in its own currency is converted to the other country’s currency using market exchange rates, one country might be inferred to have higher real GDP than the other country in one year but lower in the other; both of these inferences would fail to reflect the reality of their relative levels of production. But if one country’s GDP is converted into the other country’s currency using PPP exchange rates instead of observed market exchange rates, the false inference will not occur.

Concept

The idea originated with the School of Salamanca in the 16th century and was developed in its modern form by Gustav Cassel in 1918. The concept is based on the law of one price, where in the absence of transaction costs and official trade barriers, identical goods will have the same price in different markets when the prices are expressed in the same currency.

Another interpretation is that the difference in the rate of change in prices at home and abroad—the difference in the inflation rates—is equal to the percentage depreciation or appreciation of the exchange rate.

Deviations from parity imply differences in purchasing power of a “basket of goods” across countries, which means that for the purposes of many international comparisons, countries’ GDPs or other national income statistics need to be “PPP-adjusted” and converted into common units. The best-known purchasing power adjustment is the Geary–Khamis dollar (the “international dollar”).

The real exchange rate is then equal to the nominal exchange rate, adjusted for differences in price levels. If purchasing power parity held exactly, then the real exchange rate would always equal one. However, in practice the real exchange rates exhibit both short run and long run deviations from this value, for example due to reasons illuminated in the Balassa–Samuelson theorem.

There can be marked differences between purchasing power adjusted incomes and those converted via market exchange rates. For example, the World Bank’s World Development Indicators 2005 estimated that in 2003, one Geary-Khamis dollar was equivalent to about 1.8 Chinese yuan by purchasing power parity —considerably different from the nominal exchange rate. This discrepancy has large implications; for instance, when converted via the nominal exchange rates GDP per capita in India is about US$1,704 while on a PPP basis it is about US$3,608. At the other extreme, Denmark’s nominal GDP per capita is around US$62,100, but its PPP figure is US$37,304.

Functions

The purchasing power parity exchange rate serves two main functions. PPP exchange rates can be useful for making comparisons between countries because they stay fairly constant from day to day or week to week and only change modestly, if at all, from year to year. Second, over a period of years, exchange rates do tend to move in the general direction of the PPP exchange rate and there is some value to knowing in which direction the exchange rate is more likely to shift over the long run.

Among other uses, PPP rates facilitate international comparisons of income, as market exchange rates are often volatile, are affected by political and financial factors that do not lead to immediate changes in income and tend to systematically understate the standard of living in poor countries, due to the Balassa–Samuelson effect.

Measurement

The PPP exchange-rate calculation is controversial because of the difficulties of finding comparable baskets of goods to compare purchasing power across countries.

Estimation of purchasing power parity is complicated by the fact that countries do not simply differ in a uniform price level; rather, the difference in food prices may be greater than the difference in housing prices, while also less than the difference in entertainment prices. People in different countries typically consume different baskets of goods. It is necessary to compare the cost of baskets of goods and services using a price index. This is a difficult task because purchasing patterns and even the goods available to purchase differ across countries.

Thus, it is necessary to make adjustments for differences in the quality of goods and services. Furthermore, the basket of goods representative of one economy will vary from that of another: Americans eat more bread; Chinese more rice. Hence a PPP calculated using the US consumption as a base will differ from that calculated using China as a base. Additional statistical difficulties arise with multilateral comparisons when (as is usually the case) more than two countries are to be compared.

Various ways of averaging bilateral PPPs can provide a more stable multilateral comparison, but at the cost of distorting bilateral ones. These are all general issues of indexing; as with other price indices there is no way to reduce complexity to a single number that is equally satisfying for all purposes. Nevertheless, PPPs are typically robust in the face of the many problems that arise in using market exchange rates to make comparisons.

For example, in 2005 the price of a gallon of gasoline in Saudi Arabia was USD 0.91, and in Norway the price was USD 6.27. The significant differences in price wouldn’t contribute to accuracy in a PPP analysis, despite all of the variables that contribute to the significant differences in price. More comparisons have to be made and used as variables in the overall formulation of the PPP.

When PPP comparisons are to be made over some interval of time, proper account needs to be made of inflationary effects.

Law of One Price

Although it may seem as if PPPs and the law of one price are the same, there is a difference: the law of one price applies to individual commodities whereas PPP applies to the general price level. If the law of one price is true for all commodities then PPP is also therefore true; however, when discussing the validity of PPP, some argue that the law of one price does not need to be true exactly for PPP to be valid. If the law of one price is not true for a certain commodity, the price levels will not differ enough from the level predicted by PPP.

The purchasing power parity theory states that the exchange rate between one currency and another currency is in equilibriums when their domestic purchasing powers at that rate of exchange are equivalent.

Big Mac Index

Big Mac hamburgers, like this one from Japan, are similar worldwide.

An example of one measure of the law of one price, which underlies purchasing power parity, is the Big Mac Index, popularized by the Economist, which compares the prices of a Big Mac burger in McDonald’s restaurants in different countries.

The Big Mac Index is presumably useful because although it is based on a single consumer product that may not be typical, it is a relatively standardized product that includes input costs from a wide range of sectors in the local economy, such as agricultural commodities (beef, bread, lettuce, cheese), labor (blue and white collar), advertising, rent and real estate costs, transportation, etc.

In theory, the law of one price would hold that if, to take an example, the Canadian dollar were to be significantly overvalued relative to the U.S. dollar according to the Big Mac Index, that gap should be unsustainable because Canadians would import their Big Macs from or travel to the U.S. to consume them, thus putting upward demand pressure on the U.S. dollar by virtue of Canadians buying the U.S. dollars needed to purchase the U.S.-made Big Macs and simultaneously placing downward supply pressure on the Canadian dollar by virtue of Canadians selling their currency in order to buy those same U.S. dollars.

The alternative to this exchange rate adjustment would be an adjustment in prices, with Canadian McDonald’s stores compelled to lower prices to remain competitive.

Either way, the valuation difference should be reduced assuming perfect competition and a perfectly tradable good. In practice, of course, the Big Mac is not a perfectly tradable good and there may also be capital flows that sustain relative demand for the Canadian dollar. The difference in price may have its origins in a variety of factors besides direct input costs such as government regulations and product differentiation.

In some emerging economies, western fast food represents an expensive niche product price well above the price of traditional staples—i.e. the Big Mac is not a mainstream ‘cheap’ meal as it is in the West, but a luxury import. This relates back to the idea of product differentiation: the fact that few substitutes for the Big Mac are available confers market power on McDonald’s. Additionally, with countries like Argentina that have abundant beef resources, consumer prices in general may not be as cheap as implied by the price of a Big Mac.

The following table, based on data from The Economist’s January 2013 calculations, shows the under (−) and over (+) valuation of the local currency against the U.S. dollar in %, according to the Big Mac index. To take an example calculation, the local price of a Big Mac in Hong Kong when converted to U.S. dollars at the market exchange rate was $2.19, or 50% of the local price for a Big Mac in the U.S. of $4.37. Hence the Hong Kong dollar was deemed to be 50% undervalued relative to the U.S. dollar on a PPP basis.

Measurement Issues

In addition to methodological issues presented by the selection of a basket of goods, pp estimates can also vary based on the statistical capacity of participating countries. The International Comparison Program, which PPP estimates are based on, requires the disaggregation of national accounts into production, expenditure or (in some cases) income, and not all participating countries routinely disaggregate their data into such categories.

Some aspects of PPP comparison are theoretically impossible or unclear. For example, there is no basis for comparison between the Ethiopian laborer who lives on teff with the Thai laborer who lives on rice, because teff is not commercially available in Thailand and rice is not in Ethiopia, so the price of rice in Ethiopia or teff in Thailand cannot be determined. As a general rule, the more similar the price structures between countries, the more valid the PPP comparison.

ppp levels will also vary based on the formula used to calculate price matrices. Different possible formulas include GEKS-Fisher, Geary-Khamis, IDB, and the superlative method. Each has advantages and disadvantages.

Linking regions presents another methodological difficulty. In the 2005 ICP round, regions were compared by using a list of some 1,000 identical items for which a price could be found for 18 countries, selected so that at least two countries would be in each region. While this was superior to earlier “bridging” methods, which do not fully take into account differing quality between goods, it may serve to overstate the PPP basis of poorer countries, because the price indexing on which PPP is based will assign to poorer countries the greater weight of goods consumed in greater shares in richer countries.

Need for Adjustments to GDP

The exchange rate reflects transaction values for traded goods between countries in contrast to non-traded goods, that is, goods produced for home-country use. Also, currencies are traded for purposes other than trade in goods and services, e.g., to buy capital assets whose prices vary more than those of physical goods. Also, different interest rates, speculation, hedging or interventions by central banks can influence the foreign-exchange market.

The PPP method is used as an alternative to correct for possible statistical bias. The Penn World Table is a widely cited source of PPP adjustments, and the so-called Penn effect reflects such a systematic bias in using exchange rates to outputs among countries.

For example, if the value of the Mexican peso falls by half compared to the US dollar, the Mexican Gross Domestic Product measured in dollars will also halve. However, this exchange rate results from international trade and financial markets. It does not necessarily mean that Mexicans are poorer by a half; if incomes and prices measured in pesos stay the same, they will be no worse off assuming that imported goods are not essential to the quality of life of individuals. Measuring income in different countries using PPP exchange rates helps to avoid this problem.

exchange rates are especially useful when official exchange rates are artificially manipulated by governments. Countries with strong government control of the economy sometimes enforce official exchange rates that make their own currency artificially strong. By contrast, the currency’s black market exchange rate is artificially weak. In such cases, a PPP exchange rate is likely the most realistic basis for economic comparison.

Updating PPP rates for GDP

Since global PPP estimates —such as those provided by the ICP— are not calculated annually, but for a single year, PPP exchange rates for years other than the benchmark year need to be updated. This is done using the country’s GDP deflator. To calculate a country’s ppp exchange rate in Geary–Khamis dollars for a particular year, the calculation proceeds in the following manner:

PPP  Where PPPrate_X,i is the PPP exchange rate of country X for year i, PPPrate_X,b is the ppp exchange rate of country X for the benchmark year, PPPrate_U,b is the PPP exchange rate of the United States (US) for the benchmark year (equal to 1), GDPdef_X,i is the GDP deflator of country X for year i, GDPdef_X,b is the GDP deflator of country X for the benchmark year,

GDPdef_U,i is the GDP deflator of the US for year i, and GDPdef_U,b is the GDP deflator of the US for the benchmark year.

Difficulties

There are a number of reasons that different measures do not perfectly reflect standards of living.

Range and Quality of Goods

The goods that the currency has the “power” to purchase are a basket of goods of different types:

Local, non-tradable goods and services (like electric power) that are produced and sold domestically.

Tradable goods such as non-perishable commodities that can be sold on the international market (like diamonds).

The more that a product falls into category 1, the more that its price will be from the currency exchange rate, moving towards the PPP exchange rate. Conversely, category 2 products tend to trade close to the currency exchange rate. (See also Penn effect).

More processed and expensive products are likely to be tradable, falling into the second category, and drifting from the PPP exchange rate to the currency exchange rate. Even if the PPP “value” of the Ethiopian currency is three times stronger than the currency exchange rate, it won’t buy three times as much of internationally traded goods like steel, cars and microchips, but non-traded goods like housing, services (“haircuts”), and domestically produced crops. The relative price differential between tradables and non-tradables from high-income to low-income countries is a consequence of the Balassa–Samuelson effect and gives a big cost advantage to labour intensive production of tradable goods in low income countries (like Ethiopia), as against high income countries (like Switzerland).

The corporate cost advantage is nothing more sophisticated than access to cheaper workers, but because the pay of those workers goes farther in low-income countries than high, the relative pay differentials (inter-country) can be sustained for longer than would be the case otherwise. (This is another way of saying that the wage rate is based on average local productivity and that this is below the per capita productivity that factories selling tradable goods to international markets can achieve.) An equivalent cost benefit comes from non-traded goods that can be sourced locally (nearer the PPP-exchange rate than the nominal exchange rate in which receipts are paid). These act as a cheaper factor of production than is available to factories in richer countries.

The Bhagwati–Kravis–Lipsey view provides a somewhat different explanation from the Balassa–Samuelson theory. This view states that price levels for non-tradable are lower in poorer countries because of differences in endowment of labor and capital, not because of lower levels of productivity. Poor countries have more labor relative to capital, so marginal productivity of labor is greater in rich countries than in poor countries. Non tradable tend to be labor-intensive; therefore, because labor is less expensive in poor countries and is used mostly for non tradable, non tradable are cheaper in poor countries. Wages are high in rich countries, so non tradable are relatively more expensive.

Ppp calculations tend to over emphasis the primary sectoral contribution and under emphasis the industrial and service sectoral contributions to the economy of a nation.

Trade Barriers and non Tradable

The law of one price, the underlying mechanism behind PPP, is weakened by transport costs and governmental trade restrictions, which make it expensive to move goods between markets located in different countries. Transport costs sever the link between exchange rates and the prices of goods implied by the law of one price. As transport costs increase, the larger the range of exchange rate fluctuations. The same is true for official trade restrictions because the customs fees affect importers’ profits in the same way as shipping fees. According to Krugman and Obstfeld, “Either type of trade impediment weakens the basis of PPP by allowing the purchasing power of a given currency to differ more widely from country to country.”[4] They cite the example that a dollar in London should purchase the same goods as a dollar in Chicago, which is certainly not the case.

Non tradable are primarily services and the output of the construction industry. Non tradable also lead to deviations in PPP because the prices of non tradable are not linked internationally. The prices are determined by domestic supply and demand, and shifts in those curves lead to changes in the market basket of some goods relative to the foreign price of the same basket. If the prices of non tradable rise, the purchasing power of any given currency will fall in that country.

Departures from Free Competition

Linkages between national price levels are also weakened when trade barriers and imperfectly competitive market structures occur together. Pricing to market occurs when a firm sells the same product for different prices in different markets. This is a reflection of inter-country differences in conditions on both the demand side (e.g., virtually no demand for pork or alcohol in Islamic states) and the supply side (e.g., whether the existing market for a prospective entrant’s product features few suppliers or instead is already near-saturated).

According to Krugman and Obstfeld, this occurrence of product differentiation and segmented markets results in violations of the law of one price and absolute PPP. Over time, shifts in market structure and demand will occur, which may invalidate relative PPP

Differences in Price Level Measurement

Measurements of price levels differ from country to country. Inflation data from different countries are based on different commodity baskets; therefore, exchange rate changes do not offset official measures of inflation differences. Because it makes predictions about price changes rather than price levels, relative PPP is still a useful concept. However, change in the relative prices of basket components can cause relative PPP to fail tests that are based on official price indexes.

Global Poverty Line

The global poverty line is a worldwide count of people who live below an international poverty line, referred to as the dollar-a-day line. This line represents an average of the national poverty lines of the world’s poorest countries, expressed in international dollars. These national poverty lines are converted to international currency and the global line is converted back to local currency using the PPP exchange rates from the ICP.

Interest rate Parity

Interest rate parity is a no-arbitrage condition representing an equilibrium state under which investors will be indifferent to interest rates available on bank deposits in two countries. The fact that this condition does not always hold allows for potential opportunities to earn riskless profits from covered interest arbitrage. Two assumptions central to interest rate parity are capital mobility and perfect substitutability of domestic and foreign assets. Given foreign exchange market equilibrium, the interest rate parity condition implies that the expected return on domestic assets will equal the exchange rate-adjusted expected return on foreign currency assets. Investors cannot then earn arbitrage profits by borrowing in a country with a lower interest rate, exchanging for foreign currency, and investing in a foreign country with a higher interest rate, due to gains or losses from exchanging back to their domestic currency at maturity. Interest rate parity takes on two distinctive forms: uncovered interest rate parity refers to the parity condition in which exposure to foreign exchange risk (unanticipated changes in exchange rates) is uninhibited, whereas covered interest rate parity refers to the condition in which a forward contract has been used to cover (eliminate exposure to) exchange rate risk. Each form of the parity condition demonstrates a unique relationship with implications for the forecasting of future exchange rates: the forward exchange rate and the future spot exchange rate.

Economists have found empirical evidence that covered interest rate parity generally holds, though not with precision due to the effects of various risks, costs, taxation, and ultimate differences in liquidity. When both covered and uncovered interest rate parity hold, they expose a relationship suggesting that the forward rate is an unbiased predictor of the future spot rate.

This relationship can be employed to test whether uncovered interest rate parity holds, for which economists have found, mixed results. When uncovered interest rate parity and purchasing power parity hold together, they illuminate a relationship named real interest rate parity, which suggests that expected real interest rates represent expected adjustments in the real exchange rate. This relationship generally holds strongly over longer terms and among emerging market countries.

Assumptions

Interest rate parity rests on certain assumptions, the first being that capital is mobile- investors can readily exchange domestic assets for foreign assets. The second assumption is that assets have perfect substitutability, following from their similarities in riskiness and liquidity. Given capital mobility and perfect substitutability, investors would be expected to hold those assets offering greater returns, be they domestic or foreign assets.

However, both domestic and foreign assets are held by investors. Therefore, it must be true that no difference can exist between the returns on domestic assets and the returns on foreign assets. That is not to say that domestic investors and foreign investors will earn equivalent returns, but that a single investor on any given side would expect to earn equivalent returns from either investment decision.

A visual representation of uncovered interest rate parity holding in the foreign exchange market, such that the returns from investing domestically are equal to the returns from investing abroad.

When the no-arbitrage condition is satisfied without the use of a forward contract to hedge against exposure to exchange rate risk, interest rate parity is said to be uncovered. Risk-neutral investors will be indifferent among the available interest rates in two countries because the exchange rate between those countries is expected to adjust such that the dollar return on dollar deposits is equal to the dollar return on foreign deposits, thereby eliminating the potential for uncovered interest arbitrage profits. Uncovered interest rate parity helps explain the determination of the spot exchange rate. The following equation represents uncovered interest rate parity.

Where

is the expected future spot exchange rate at time t + k k is the number of periods into the future from time t

S_t is the current spot exchange rate at time t

i_$ is the interest rate in the US

_c is the interest rate in a foreign country or currency area (for this example, following

a US perspective, it is the interest rate available in the Eurozone)

The dollar return on dollar deposits, , is shown to be equal to the dollar return on euro deposits,

Approximation

Uncovered interest rate parity asserts that an investor with dollar deposits will earn the interest rate available on dollar deposits, while an investor holding euro deposits will earn the interest rate available in the euro-zone, but also a potential gain or loss on euros depending on the rate of appreciation or depreciation of the euro against the dollar. Economists have extrapolated a useful approximation of uncovered interest rate parity that follows intuitively from these assumptions.

If uncovered interest rate parity holds, such that an investor is indifferent between dollar versus euro deposits, then any excess return on euro deposits must be offset by some expected loss from depreciation of the euro against the dollar. Conversely, some shortfall in return on euro deposits must be offset by some expected gain from appreciation of the euro against the dollar. The following equation represents the uncovered interest rate parity approximation.

Where

 is the change in the expected future spot exchange rate

 is the expected rate of depreciation of the dollar

A more universal way of stating the approximation is “the home interest rate equals the foreign interest rate plus the expected rate of depreciation of the home currency.”

A visual representation of covered interest rate parity holding in the foreign exchange market such that the returns from investing domestically are equal to the returns from investing abroad.

When the no-arbitrage condition is satisfied with the use of a forward contract to hedge against exposure to exchange rate risk, interest rate parity is said to be covered. Investors will still be indifferent among the available interest rates in two countries because the forward exchange rate sustains equilibrium such that the dollar return on dollar deposits is equal to the dollar return on foreign deposit, thereby eliminating the potential for covered interest arbitrage profits. Furthermore, covered interest rate parity helps explain the determination of the forward exchange rate. The following equation represents covered interest rate parity.

Where

 is the forward exchange rate at time t

The dollar return on dollar deposits, is shown to be equal to the dollar return on euro deposits,

Empirical Evidence

Covered interest rate parity (CIRP) is found to hold when there is open capital mobility and limited capital controls, and this finding is confirmed for all currencies freely traded in the present-day. One such example is when the United Kingdom and Germany abolished capital controls between 1979 and 1981. Maurice Obstfeld and Alan Taylor calculated hypothetical profits as implied by the expression of a potential inequality in the CIRP equation (meaning a difference in returns on domestic versus foreign assets) during the 1960s and 1970s, which would have constituted arbitrage opportunities if not for the prevalence of capital controls.

However, given financial liberalization and resulting capital mobility, arbitrage temporarily became possible until equilibrium was restored. Since the abolition of capital controls in the United Kingdom and Germany, potential arbitrage profits have been near zero. Factoring in transaction costs arising from fees and other regulations, arbitrage opportunities are fleeting or nonexistent when such costs exceed deviations from parity. While CIRP generally holds, it does not hold with precision due to the presence of transaction costs, political risks and tax implications for interest earnings versus gains from foreign exchange, and differences in the liquidity of domestic versus foreign assets. Researchers found evidence that significant deviations from CIRP during the onset of the global financial crisis in 2007 and 2008 were driven by concerns over risk posed by counter parties to banks and financial institutions in Europe and the US in the foreign exchange swap market. The European Central Bank’s efforts to provide US dollar liquidity in the foreign exchange swap market, along with similar efforts by the Federal Reserve, had a moderating impact on CIRP deviations between the dollar and the euro. Such a scenario was found to be reminiscent of deviations from CIRP during the 1990s driven by struggling Japanese banks which looked toward foreign exchange swap markets to try and acquire dollars to bolster their creditworthiness.

When both covered and uncovered interest rate parity (UIRP) hold, such a condition sheds light on a noteworthy relationship between the forward and expected future spot exchange rates, as demonstrated below.

Dividing the equation for UIRP by the equation for CIRP yields the following equation:

which can be rewritten as:

This equation represents the unbiasedness hypothesis, which states that the forward exchange rate is an unbiased predictor of the future spot exchange rate. Given strong evidence that CIRP holds, the forward rate unbiasedness hypothesis can serve as a test to determine whether UIRP holds (in order for the forward rate and spot rate to be equal, both CIRP and UIRP conditions must hold). Evidence for the validity and accuracy of the unbiasedness hypothesis, particularly evidence for co integration between the forward rate and future spot rate, is mixed as researchers have published numerous papers demonstrating both empirical support and empirical failure of the hypothesis.

UIRP is found to have some empirical support in tests for correlation between expected rates of currency depreciation and the forward premium or discount. Evidence suggests that whether UIRP holds depends on the currency examined, and deviations from UIRP have been found to be less substantial when examining longer time horizons. Some studies of monetary policy have offered explanations for why UIRP fails empirically. Researchers demonstrated that if a central bank manages interest rate spreads in strong response to the previous period’s spreads, that interest rate spreads had negative coefficients in regression tests of UIRP. Another study which setup a model wherein the central bank’s monetary policy responds to exogenous shocks, that the central bank’s smoothing of interest rates can explain empirical failures of UIRP. A study of central bank interventions on the US dollar and Deutsche mark found only limited evidence of any substantial effect on deviations from UIRP. UIRP has been found to hold over very small spans of time (covering only a number of hours) with a high frequency of bilateral exchange rate data. Tests of UIRP for economies experiencing institutional regime changes, using monthly exchange rate data for the US dollar versus the Deutsche mark and the Spanish peseta versus the British pound, have found some evidence that UIRP held when US and German regime changes were volatile, and held between Spain and the United Kingdom particularly after Spain joined the European Union in 1986 and began liberalizing capital mobility.

Real Interest Rate Parity

When both UIRP (particularly in its approximation form) and purchasing power parity (PPP) hold, the two parity conditions together reveal a relationship among expected real interest rates, wherein changes in expected real interest rates reflect expected changes in the real exchange rate. This condition is known as real interest rate parity (RIRP) and is related to the international Fisher effect. The following equations demonstrate how to derive the RIRP equation.

Where  represents inflation.

If the above conditions hold, then they can be combined and rearranged as the following:

RIRP rests on several assumptions, including efficient markets, no country risk premia, and zero change in the expected real exchange rate. The parity condition suggests that real interest rates will equalize between countries and that capital mobility will result in capital flows that eliminate opportunities for arbitrage. There exists strong evidence that RIRP holds tightly among emerging markets in Asia and also Japan. The half-life period of deviations from RIRP have been examined by researchers and found to be roughly six or seven months, but between two and three months for certain countries. Such variation in the half-lives of deviations may be reflective of differences in the degree of financial integration among the country groups analyzed. RIRP does not hold over short time horizons, but empirical evidence has demonstrated that it generally holds well across long time horizons of five to ten years.