Time Series Econometrics Course Work 2

Time-Series Econometrics Coursework 2 Name: Alejandro Calaf Fernndez URN: 6131104

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Time Series Econometrics Course Work 2

In this paper, the relationship between exchange rates of two countries and their respective price levels will be

scrutinised. Understanding the adjustment of exchange rate is a crucial for policy makers, since countries with both

fixed and floating exchange rates are interested in knowing what the equilibrium exchange rate would be and what

variation in nominal and real exchange rates they should anticipate.

The natural extension of LOP1, on the economy as a whole, is the Purchasing Power Parity (PPP). It states that the

nominal exchange rate between two currencies should be equal to the ratio of aggregate price levels between two

countries, so that a unit of currency from one country will have the same purchasing power in a foreign country2. In

other words, movements in exchange rates simply denote movements in the price levels (a basket of goods) of two

economies, excluding transaction costs (arising from transport costs, tariffs and nontariff barriers). This relationship

can be described by the following equations:

=

+

= +

For the purpose of model building, it is a necessary condition to test if the series are stationary I(0) or non-stationary

I(d). By inspecting the series, suspicion arises that the series for exchange rates and price levels for both domestic and

foreign countries might be integrated of some order since there is no evidence of mean reversion or time-

independent variance.

Figure 1: Plot of nominal exchange rate series Figure 2: Plot of domestic price level series Figure 3: Plot of foreign price level series

To formally test for the stationarity of these series several tests are performed, namely Augmented Dickey-Fuller

(ADF) test3 and Phillips-Perron (PP) test4.

1 The Law of One Price (LOP) = theory stating that price of internationally traded goods should be equal anywhere in

the world providing that the price is expressed in a common currency. Results from arbitrage opportunities, where

riskless profit could be obtained by acquiring a product where the price is low and selling it in a location with a higher

price.

2Alan M. Taylor, and Mark P. Taylor. "The Purchasing Power Parity Debate." 3 We use the Augmented Dickey-Fuller (ADF) test instead of Dickey Fuller test since the error term in the latter one is autocorrelated. A parametric way to correct for serial correlation in error term is using ADF test. 4 Phillips-Perron (PP) test corrects the serial correlation in the error term by modifying the test statistic using a non-parametric approach

et =Exchange Rate; pt =Price level domestic country; pft =Price level foreign country

After logarithm transformation

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CALAFFERNANDEZ1E

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CALAFFERNANDEZ2P

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CALAFFERNANDEZ3PF


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Figure 4: ADF test with trend and intercept on nominal exchange rates Figure 5: ADF test with intercept on nominal exchange rates

All of the three series above behave as a random walk without a drift, implying that while performing a unit root test

a trend should not be included. However, to test if a trend should be included, perform the ADF test with intercept

and trend and check for the significance of the trend coefficient.

In the estimation output in figure 4 the coefficient for @trend(1) is highly insignificant implying that a trend should

not be included in the unit root test for stationarity

In the estimation output in figure 5, the coefficient for the intercept (C) is significant implying that it should be

included for performing an ADF test. The p-value obtained, 0.0026, forces us to reject the null-hypothesis and accept

the alternative hypothesis at a 99% level of confidence. Where the hypotheses are as follow

H0: series has a unit root = series is non stationary

H1: series has no unit root = series is stationary

The PP test shares the same hypotheses as the ADF test. In figure 6 the results from this test on nominal exchange

rates including and intercept term are shown. The P-value attained makes us reject the null hypothesis and accept the

alternative hypothesis at a 99% level of confidence5.

5 It should be noted the both the ADF and PP test are feeble test in the sense that there is a probability of rejecting the null hypothesis when in fact it is true. Moreover, both test share the disadvantage of sensitivity to structural breaks (which can be observed in the plot of the series Figure 1, 2 and 3) and lack of power in small samples. In conclusion, it shouldnt be ruled out that the series are non-stationary.


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Figure 6: PP test with intercept on nominal exchange rates These tests for stationarity are repeated for the other two variables

present in the model ( and ). As it can be observed from figure 2

and 3, the series seems to display neither a deterministic trend nor

do they seem to be fluctuating around a non-zero mean. This implies

that the aforementioned tests for stationarity should not include a

trend but should include an intercept term. The results for the 3 tests

of stationarity are summarised in table 1 and 2 respectively.

Table 1: Unit root test for calaf2p series (domestic price index)

Table 2: Unit root test for calaf3pf series (foreign price index)

In appendix 3, it can be seen that if an ADF test is performed on the first difference of both series (calaf2p and

calaf3pf), they are rendered stationary. This implies that both series are integrated of order 1 (I(1)) and that the series

are a difference stationarity process (DSP).

A central question for continuing the analysis, is finding whether nominal exchange rates are correlated with past

values of the foreign price index series. This can be formally tested using the Granger Causality test. To perform this

test, a necessary condition is that the series being scrutinised have to display a stationary process. Once results are

obtained, there can be unidirectional and bidirectional causality or independence. Essentially, Granger Causality


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measures whether current and past values of the foreign price index () aid to forecast future values of nominal

exchange rates (), or vice versa.

As abovementioned, the series calaf3pf (log of foreign price index) are I(1), this means that if the first difference is

taken on the series it will be rendered stationary. As a result, a new series will be obtained which would be equal to

the change in foreign price index or equivalently the foreign inflation rate. Although this procedure is necessary, by

differencing the series, useful long run information about the causal relationship between the variables is

condemned.

A Vector Autoregressive Model (VAR) has to be estimated to execute a Granger Causality test. A tool used to select

the correct lag length of a given VAR model, is to compare the Akaike, Schwarz and Hannan-Quinn criterion for

models of different lags and select the model that minimises these values. From the results obtained in Appendix 4,

the conclusion is that a VAR(1) model (Appendix 5) should be estimated to carry out the Granger Causality test. In

other words, the model being estimated is of this sort:

Figure 8: Granger Causality test on VAR(1) model

The null and alternative hypotheses are:

H0: 12 = 0 = No causality from 1 to

H1: 12 0 = Causality from 1 to Or:

H0: 21 = 0 = No causality from 1 to

H1: 21 0 = Causality from 1 to

From the test output in figure 8, it can be concluded that there exists some

degree of causality from nominal exchange rates to changes in the foreign price

index, since 21 is significantly different from 0 (looking at the p-value

0.0130) . Furthermore, the null hypothesis of 12 = 0 is failed to be rejected (p-value 0.6530), ergo concluding that

current values of the nominal exchange rate will not be affected by past values of . These results can be stressed

by looking at the impulse response functions which are used to produce the time path of the dependant variable in a

VAR model to shocks from all the explanatory variables.

= 1

3 = 3 31

(

) = (12

) + (11 1221 22

) + ( 11

) + (12

)


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Figure 9: Impulse Response functions for VAR(1) model

From these impulse response functions, it can be seen in the

second panel that the response of nominal exchange rates

(calaf1e) to shocks in the foreign countrys inflation rate

(dcalaf3pf) is 0. Moreover the confidence bands suggest that a

shock to dcalaf3pf will originate a reaction from the nominal

exchange rates not significantly different from 0. These results,

of a flat impulse response function, are consistent from the

ones obtained in the Granger causality test. In the third panel it

can be seen that a shock to nominal exchange rate will result in

a negative foreign countrys inflation rate during the first

period, which will revert to a positive response in the second

period and gradually revert to 0. From the confidence band it

can be concluded that the response of dpftto a shock in et1will

be significantly different from 0. A positive shock to the nominal exchange rate will depreciate the foreign currency

and ergo reducing price levels (in terms of the domestic currency) in the short run, making foreign goods more

attractive. Through trade, the demand for foreign goods will increase causing an increase in the price level (in terms of

the domestic currency) of the foreign country in the medium term, which will in turn reduce the appeal of foreign

goods internationally.

An important part for testing the validity of the PPP hypothesis, knowing that PPP is a long run condition, is evaluating

the long run relationships between the series being examined, namely price indices and nominal exchange rates.

However, rendering the series stationary through differentiating will jeopardize important long-run information.

The theory states that any linear combination of non-stationary variables will also be non-stationary. Nevertheless, if

the residuals in said linear combination are proven to be stationary it would imply that the series are cointegrated6. It

indicates that the series share similar stochastic trends and since the difference,ut, is stationary they are never

expected to drift too far away from each other. Ergo, there exists a long run relationship between the variables. A

necessary condition for cointegration is that two variables should be integrated of the same order (Appendix 3). To

test for this long run relationship three methods can be used.

Imposing no restriction will lead to estimating a model as such:

= + From this regression the residuals are obtained

= + Implying that the cointegrating vector will be [1, -, ]

6 This test is formally carried out using the Engle-Granger test for cointegration; however a limitation of Eviews is that residuals being used for the stationarity test are taken from an OLS regression instead of a DOLS or FMOLS regression. Hence, the test is carried out manually by saving up the residuals from said regressions and testing their stationarity with an ADF test. The critical values used to compare the t statistic are found in the Engle-Granger table for residual-based tests.


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Figure 10: Estimation Output for unrestricted model using OLS

By eyeballing the graph of residuals (Appendix 6), it can be conclude

that in the ADF test neither an intercept nor a trend term should be

included. The null and alternative hypotheses of this test are the

following:

H0: Residual has unit root = Series are not cointegrated

H1: Residual has no unit root = Series are cointegrated

Comparing the t statistic obtained in the ADF test (Appendix 7), -

5.605311, with the critical value at a 95% level of confidence, -

3.7429, the null hypothesis is rejected. Ergo, concluding that the residuals are stationary and consequently the series

are cointegrated. The static OLS estimation of the unrestricted model has managed to capture the long run

relationship between the variables. The interpretation is that a 1% increase in the domestic price index will lead to a

1.00798% increase in the nominal exchange rate (in the long-run) and a 1% increase in the foreign price index will

cause a 0.939708% decrease in the nominal exchange rate (in the long-run). Although the OLS estimation is super

consistent7, minimising the variance of residuals, it is biased in small samples.

Figure 11: Estimation Output for unrestricted model using DOLS

A solution, would be estimating the above equation using Dynamic OLS. It corrects for the endogeneity bias by augmenting the cointegrating regression with lags and leads8. Once again the graph of residuals (Appendix 8) displays no evidence that a trend or an intercept term should be included in the ADF test for stationarity (Appendix 9). By comparing the t-statistic, -5.453861, with the critical value at 95% level of confidence, -3.7429, the null hypothesis is once again rejected. Concluding that the residuals are not stationary, thus the series are cointegrated and display a long run relationship. The interpretation of the estimation output in Figure 11 is quite similar to the previous one. A 1% increase in the foreign price index will lead to a 1.009270% increase in the nominal exchange rate (in the long run) and a 1% increase in the foreign price index will trigger a 0.916532%

decrease in the nominal exchange rate (in the long run). Figure 12: Estimation Output for unrestricted model using FMOLS

Finally, Fully Modified OLS also corrects for the endogeneity bias in a

non-parametric manner, allowing it to be an unbiased and efficient

estimator. The same process as before is carried out where the

residuals are obtained from the FMOLS regression (Appendix 10) and

perform an ADF test (Appendix 11) to test the stationarity of the

residuals. The same conclusion of stationarity in the residuals and

cointegration in the series, ergo displaying a long run relationship, is

attained when the t-statistic, -5.742710, is compared with the critical

value at 95% level of confidence, -3.7429. In figure 12, the values can

be interpreted as a 1% increase in the domestic price index will cause

a 1.015556% increase in the nominal exchange rate, whilst a 1% increase in the foreign price index will trigger a

0.914905 decrease in the nominal exchange rate (in the long run).

7 The estimated value of and converge to the true parameters at a rate of T-1 (faster) instead of T-1/2 with a consistent estimator. 8 Lags and leads were selected using Schwarz Criterion since it is more lenient with more parsimonious models.


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In conclusion, these long run relationships are fairly close to the theoretical long run relationship that the PPP hypothesis predicts (1% increase in domestic price index leading to a 1% increase in nominal exchange rate and 1% increase in foreign price index causing a 1% decrease in nominal exchange rates), hence the long run PPP hypothesis is proven using the unrestricted formulation. A semi restricted formulation of the PPP hypothesis can be used to test for the existence of long run relationship amongst the variables. The following should be carried out. Generate series = +

= Where the theoretical cointegrating vector would be [1]

Table 3: Semi-restricted formulation of PPP hypothesis for testing long-run relationship

The above equation is regressed using static OLS,

DOLS and FMOLS. The residuals series from these

regressions is obtained and a test for the

stationarity of the residuals is executed. As

explained before, if residuals are found to not have

a unit root, the series will be cointegrated ergo

implying a long run relationship. The results are

summarised in table 3. Once again, long run PPP

can be confirmed since the theoretical

cointegrating vector is fairly close to the estimated cointegrating vector using static OLS, DOLS and FMOLS [1.094698],

[1.055947] and [1.067874] respectively, since constant is insignificant. The interpretation of the coefficients of are

that a 1% increase in the domestic price index will lead to a 1.094698%, 1.055947% and 1.067874% respective

increase in the foreign price index, in terms of the domestic currency.

A restricted formulation of the PPP hypothesis would be equivalent to generating the following series = +

A test on the stationarity of the series is carried out. It is observed in figure 13 that the series has no clear trend and does not wander around a non-zero mean. Thus, proceeding to an ADF test including a constant but without a trend term.

Figure 13: Plot of generated series Figure 14: ADF for stationarity on generated series

From figure 14 it is concluded that the series is stationary,

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UT


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thus once more the long run relationship in the PPP hypothesis is proven. More over the series could be interpreted as the real exchange rate between the two countries. I decide to continue my analysis using the unrestricted formulation of the PPP hypothesis since it allows for a clearer interpretation of the coefficients.

The following step in our analysis is to construct a model that describes the long run and short run behaviour of the

series, as well as the speed of adjustment at which the dependant variable returns to equilibrium when there is a

shock in the explanatory variables and . A model analysing these dynamics is the Error Correction Model

(ECM). The long run relationship will be captured by the equilibrium error, acting as a gravitational pull after a short

run shock, bringing the series back to its equilibrium. The standard OLS is an efficient estimation for this model;

however it requires variables to be stationary. Subsequently, the variables will have to be differenced to render them

stationary.

ECM using static OLS residuals

To estimate the ECM, the first difference of the series , , and to make sure that the series is

stationary, is taken. In the ECM estimation (Appendix 18) 8 difference lags of the series and are

included to make sure all the correlation from residuals is captured. Proceeding, then, to gradually eliminate

the difference lags with higher insignificant coefficients rendering the model to a more parsimonious form.

Caution needs to be taken in this process since it is key to obtain a significant error correction coefficient

(resid_unrestrict_ols(-1)) implying that there is cointegration. Ergo, the following ECM was obtained

(Appendix 19):

= 0 + (__)1 + 0 + 1 +

The coefficient of is -0.269950, implying that 26.995% of a shock in the previous period is corrected in the

current period. Ergo it represents the speed of adjustment in the model. Assuming quarterly data, this means

that it will take roughly 4 quarters (3.704 quarters) to fully adjust from an innovation. The short run effects of

the model are that a 1 percentage point (pp) increase in will lead to a 1.035193pp increase in whilst a

1pp increase in will result in a 1.038185pp decrease in .

ECM using DOLS residuals

The same procedure as before is employed to obtain the ECM using DOLS residuals. The analysis and

interpretation of the coefficients are summarised in table 4.

= 0 + (__)1 + 0 + 1 +

Table 4: Interpretation and explanation of coefficients in ECM using DOLS residuals.


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ECM using FMOLS residuals

Just as before, the same procedure to obtain the ECM using FMOLS residuals was followed. The analysis and

interpretation of coefficients is summarised in table 5.

= 0 + (__)1 + 0 + 1 +

Table 5: Interpretation and explanation of coefficients in ECM using FMOLS residuals.

Obtaining a good forecast model is crucial for any policymaker and economic agent. As explained previously, countries

with different exchange rate regimes, whether these are floating or fixed exchange rates, are interested in obtaining

accurate forecast of the nominal exchange rates so that their monetary policy could be adjusted accordingly.

Moreover the import, export and financial industries rely heavily on currency exchange rates; hence it is in their

interest to obtain precise predictions on nominal exchange rates.

An AR(p) type model is developed to be compared in terms of in sample predictability and out of sample forecasting

ability with the 3 ECMs developed so far. The procedure used to obtain the AR(2) model is descried in Appendix 24.

= 0 + 11 + 22

A 10-step-ahead forecast is obtained in all 4 models by removing the last 10 observations in the sample and carrying

out again the regression with the reduced sample size. Ergo, the previous models are estimated with a reduced

sample size ranging from observation 1 until observation 190, and the forecast is obtained from observation 191 until

the end of the sample, observation 200.

Figure 15: Forecast ECM using OLS residuals

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DCALAF1EF 2 S.E.

Forecast: DCALAF1EF

Actual: DCALAF1E

Forecast sample: 191 200

Included observations: 10

Root Mean Squared Error 1.029700

Mean Absolute Error 0.906036

Mean Abs. Percent Error 77.57305

Theil Inequality Coefficient 0.203536

Bias Proportion 0.001017

Variance Proportion 0.005839

Covariance Proportion 0.993144


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Figure 16: Forecast ECM using DOLS residuals

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DCALAF1EF 2 S.E.

Forecast: DCALAF1EF

Actual: DCALAF1E










Figure 17: Forecast ECM using FMOLS residuals

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DCALAF1EF 2 S.E.

Forecast: DCALAF1EF

Actual: DCALAF1E










Figure 18: Forecast AR(2) model

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DCALAF1EF 2 S.E.

Forecast: DCALAF1EF

Actual: DCALAF1E










It is observed that both the root mean squared error (RMSE) and the mean absolute error (MAE) are relatively high

for all 4 models suggesting that the forecast are not close to the observed values. Furthermore, the mean absolute

percentage errors (MAPE) for all the models are once again relatively high. In the AR(2) model, this high value of

MAPE can be explained due to failure in the model to account for the variability in the out-of sample data. In the

ECMs, these values are marginally lower suggesting that ECMs perform better in their forecasting ability.

The Theils inequality coefficients for the ECMs are somewhat close to 0 suggesting that our forecasts obtained

through these models are reliable. Nevertheless, the Theils inequality coefficient obtained through the AR(2) process

is fairly close to 1, indicating that a nave forecast will perform just as well.

Finally, both the bias and variance proportion in the ECMs are quite low, implying that both the forecasted mean and

variance respectively are fairly close to the observed mean and variance. Conversely, the bias and variance proportion


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obtained in the forecast of the AR(2) process are higher, suggesting that this model underperforms in terms of out-of-

sample forecasting ability.

In table 6 and 7 a comparison is shown on the performance in terms of insample predictability and out-of-sample

forecasting ability.

Table 6: Model comparison in terms of in-sample predictability

The model preferred in terms of fitting past data is the ECM using FMOLS residuals since it marginally minimises all of

the criteria above.

Table 7: Model comparison in terms of out-of-sample forecasting performance

Nonetheless, in terms of forecasting capacity, it is shown that the model that marginally mimeses most of the criteria

used to measure the forecasting aptitude is the ECM using OLS residuals. This result is consistent with our

expectation, since if and are cointegrated (as it has been shown) OLS is super-consistent. Implying that the

estimated parameter for converges to the true value of at a much faster rate.

The preferred model in terms of both out-of-sample forecasting aptitude and in-sample predictability is the ECM using

FMOLS residuals since it manages to correct the endogeneity bias and serial correlation present in the ECM using OLS

residuals.

Furthermore, it is important to note that neither of the ECMs forecasts have managed to yield a forecast for the

nominal exchange rate close to 1, as the theory predicts. This can be explained by the omission of important variables

in the model such as transportation cost or barriers of trade.

In conclusion, the absolute PPP hypothesis has been proven using the data available. The non stationarity of the series

was tested using unit root tests to both avoid meaningless results in Granger Causality tests and to carry out

cointegration tests when all the series were found to be I(1). Cointegration tests were executed on different

formulations of the absolute PPP hypothesis, where the long run relationship was proven since cointegration vectors

were found to be close to the theoretical ones that the theory predicts. Moreover, having found that the series were

cointegrated, the short run dynamics of the series and the adjustment towards the long run equilibrium was tested by

generating ECMs. Finally, the different ECMs were compared against an AR model in terms of in-sample predictability

and out-of-sample forecasting ability, and it was concluded that the ECMs outperformed the AR model in both.


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Appendices

Appendix 1

ADF test with intercept without trend on calaf2p PP test with intercept without trend on calaf2p

Appendix 2

ADF test with intercept without trend on calaf3pf PP test with intercept without trend on calaf3pf


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Appendix 3

1st difference ADF test without intercept or trend on calaf2p 1st difference ADF test without intercept or trend on calaf3pf

Appendix 4 Appendix 5

Lag selection criteria for VAR model Estimation Output for VAR(1) model


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Plot of residuals obtained through unrestricted estimation using OLS ADF test on residuals of unrestricted model using OLS

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RESID_UNRESTRICT_OLS


Plot of residuals obtained through unrestricted estimation using DOLS ADF test on residuals of unrestricted model using DOLS

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Plot of residuals obtained through unrestricted estimation using FMOLS ADF test on residuals of unrestricted model using FMOLS

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Plot of residuals obtained through semi-restricted estimation using OLS ADF test on residuals of semi-restricted model using OLS

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Plot of residuals obtained through semi-restricted estimation using DOLS ADF test on residuals of semi-restricted model using DOLS

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Plot of residuals obtained through semi-restricted estimation using FMOLS ADF test on residuals of semi-restricted model using FMOLS

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8 Difference lag ECM using OLS residuals Final ECM using OLS residuals


8 Difference lag ECM using DOLS residuals Final ECM using DOLS residuals


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8 Difference lag ECM using FMOLS residuals Final ECM using FMOLS residuals

Appendix 24

Correlogram of Correlogram of residuals in AR(2) process


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Estimation output in AR(2) process


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