Testing Purchasing Power Parity
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Published: Tue, 17 Oct 2017
Within this paper a combination of univariate models (which assume the value of the exchange rate is based on its past values) and multivariate models (which will consider the trend in domestic and foreign prices) will be used to test the Purchasing Power Parity hypothesis. The data used will be the CPI for Japan and Sweden along with the exchange rate for the Japanese Yen and Swedish Krona. The paper will aim to find the best univariate and multivariate models and then compare their performance to the actual data.
For the univariate analysis the log of the exchange rate (e_{t}) and the respective growth rate (âˆ†e_{t}) will be considered for the period of July 1997 – September 2013, the data for October 2013 – September 2014 will be emitted and used for testing the models.
1. Estimate appropriate ARMA models for the log of the nominal exchange e
And for the respective growth rate (âˆ†e), carefully explaining how your models were selected.
From initial inspection of the graph of e_{t} the data seems to follow a random walk but there is some evidence that it reverts to a fixed mean which could be an example of an AR(1) process with the coefficient close to 1. The histogram does not depict a normal distribution as there are a number of peaks, this is supported by the JarqueBera test and we can conclude we do not have a normal distribution as the Pvalue is insignificant at the 1% significance level and therefore we reject the null hypothesis of normally distributed errors.
From the correlogram it appears that the process is an AR(1) as the autocorrelation dies down gradually and the partial correlation dies out after 1 lag. Other models will also be tested to ascertain if there are better models available. From this testing the best model found based on the information criteria was an AR(1) AR(4) with the lags being significant at the 5% significance level. The Pvalue for the 4^{th} lag in the AR(1) AR(4) model was only just below this significance level, which could suggest the AR(1) model will actually perform better when forecasting, this model was also significant at the 5% significance level and will also be used as a model with less lags may perform better and the information criteria were only slightly worse (see appendix i for a list of regression outputs and significance levels).
Model 
AIC 
SC 
HQ 

AR(1) 




AR(1) AR(4) 



It can be shown that both of these models are stationary which you would expect when comparing the exchange rates between two countries whose economies have followed a similar path.
Following the same process for âˆ†e for e it appears that the data is white noise with a large amount of noise around 07/08, this could be as a consequence of the recession at that time. The Pvalue for the JarqueBera test is insignificant at the 1% significance level and therefore we reject the null hypothesis of normally distributed errors. The mean is close to zero but there is negative skewness and high kurtosis.
Examining the correlogram shows no distinct signs of Autocorrelation or Partial correlation which could imply an ARMA model. Using trial and error and examining the information criteria as well as the significance of the lags; the best two models were as below, no other models were significant at the 5% significance level with respect to variables (see appendix ii for a list of regression outputs and significance levels).
Model 
AIC 
SC 
HQ 

AR(1) MA(1) 




Constant 



This suggests that an AR(1) MA(1) model is best but also that using a model which assumes that âˆ†_{et } is constant will also work as a good model for the exchange rate.
2. Obtain 12step ahead forecasts for the mean of e using both modelling approaches from 1. and assess their forecasting ability.
The forecasted values were calculated 12 steps ahead for each model and then compared to the actual results using the RMSE, MAE, MAPE and U. For the two models for e_{t} AR(1) and AR(1) AR(4) the results suggest that the AR(1) model performed better at forecasting due to the lower RMSE, MAE, MAPE and U values. The forecasts for âˆ†e_{t} were converted to forecasts of e_{t} so that the results were comparable.
For the two models using âˆ†e_{t} it can be seen that the constant model performs better as a forecast for future exchange rates between Japan and Sweden. This suggests that the exchange rate will continue to adjust at the same rate as in the last period accounted for (2013m09). We can conclude that the constant model is the best forecast method for the logged exchange rates between Japan and Sweden as all 4 indicators are smallest for this model. It is however unlikely in practice for an exchange rate to grow at a constant rate as usually exchange rates fluctuate largely as a result of shocks to the respective economies. It perhaps suggests that the other models are not largely successful at forecasting changes in the exchange rate instead of suggesting strength of the model with no allowance for past lags. This could be due to the other models overestimating fluctuations in the exchange rates. This could be due to the presence of ARCH effects.
Forecast 

Model 
RMSE 
MAE 
MAPE 
U 
e AR(1) AR(4) 
0.0533860 
0.0476250 
1.7314010 
0.0098290 
e AR(1) 
0.0439990 
0.0368120 
1.3360190 
0.0080850 
âˆ†e AR(1) MA (1) 
0.0350707 
0.0276425 
1.0021864 
0.0064299 
âˆ†e constant 
0.03205544 
0.02585021 
0.93857607 
0.00586893 
3. Estimate a GARCHtype model for e, carefully explaining how your preferred model was selected. Test for the presence of ARCH effects in the residuals.
For all of the 4 above models when viewing the squared residuals for each of the previously tested models it can be seen that the p values are less than 0.1 and when conducting the ARCH heteroskedacity test and examining the Pvalue of the chisquared statistic the value is always close to 0 therefore for all models we reject the null hypothesis of no ARCH effects and conclude that the use of an ARCH or GARCH model is needed. As earlier mentioned e_{t} and âˆ†e_{t} are not normally distributed and so the tdistribution will be used.
Model 
Pvalue of chisquared stat 
Pvalue squared residuals HQ 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 

e_{t} AR(1) AR(4) 
0.0353 
0.002 
0.001 
0.000 
0.000 
0.001 
0.002 
0.003 
0.006 
0.008 
0.013 

e_{t} AR(1) 
0.0030 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.001 

âˆ†e_{t} constant 
0.0063 
0.006 
0.000 
0.001 
0.000 
0.000 
0.000 
0.000 
0.000 
0.001 
0.002 
0.002 
0.003 
âˆ†e_{t} AR(1) MA(1) 
0.0311 
0.001 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.001 
0.001 
0.002 
Initially the AR(1) AR(4) model for e_{t} was tested and the pvalue for AR(4) was insignificant, therefore this was dropped and an AR(1) model was tested. This produced significant results for all variables and eliminated arch effects as shown by the pvalues of squared residuals being greater than 0.1 and the ARCH heteroskedacity test producing a significant pvalue. This model was also tested with higher order ARCH/GARCH but no better models were found as shown by the information criteria below. Therefore the AR (1) GARCH (1,1) model will be used going forward. (Full results can be seen in appendix iii).
Model 
AIC 
SC 
HQ 
AR(1) AR(4) GARCH (1,1) 
3.659711 
3.54997 
3.61537 
AR(1) GARCH (1,1) 
3.647996 
3.55485 
3.61037 
AR(1) GARCH (0,1) 
3.629104 
3.55148 
3.59775 
AR(1) ARCH (1) 
3.625353 
3.54773 
3.594 
AR(1) GARCH (2,1) 
3.643651 
3.53497 
3.59976 
AR(1) GARCH (1,2) 
3.643243 
3.53457 
3.59935 
The same process was used for the two models of âˆ†e and the information criteria suggest a AR(1) MA(1) GARCH (1,1) is best. This also removed ARCH effects as shown by the pvalues of the squared residuals all being greater than 0.1 the ARCH heteroskedacity test producing a significant pvalue (see appendix iv for full results). Going forward the AR(1) GARCH (1,1) for e_{t} will be used and the AR(1) MA(1) GARCH (1,1) for âˆ†e_{t.}
Model 
AIC 
SC 
HQ 
Constant GARCH (1,1) 
3.648715 
3.57109 
3.61736 
Constant GARCH (2,1) 
3.646597 
3.55345 
3.60897 
Constant GARCH (1,2) 
3.64412 
3.55097 
3.6065 
Constant GARCH (0,1) 
3.627903 
3.5658 
3.60282 
Constant ARCH (1) 
3.630404 
3.5683 
3.60532 
AR(1) MA(1) GARCH (1,1) 
3.665609 
3.55658 
3.62157 
AR(1) MA(1) GARCH (2,1) 
3.665693 
3.54109 
3.61536 
AR(1) MA(1) GARCH (1,2) 
3.661519 
3.53691 
3.61118 
AR(1) MA(1) GARCH (0,1) 
3.645766 
3.55231 
3.60802 
AR(1) MA(1) ARCH (1) 
3.65517 
3.56172 
3.61742 
4. Test if there is any evidence of:
i) a link between volatility and nominal exchange rate movements.
By running a GARCH model and altering the arch in mean to use the standard deviation or log of variance, a test for a link between volatility and nominal exchange rate movements can be conducted. If the model is significant and the information criteria improve this shows there is a link between volatility and nominal exchange rate movements. This will be tested for both the AR(1) MA(1) GARCH (1,1) for âˆ†e_{t} and the AR(1) GARCH (1,1) for e_{t}.
e_{t} AR(1) GARCH (1,1) 

AIC 
SC 
HQ 
Pvalue of the regressor 

none 
3.647996 
3.554845 
3.610371 
– 
Std. dev. 
3.638826 
3.530149 
3.594930 
0.9758 
Log (var) 
3.639104 
3.530428 
3.595208 
0.7914 
âˆ†e_{t} AR(1) MA(1) GARCH (1,1) 

AIC 
SC 
HQ 
Pvalue of the regressor 

none 
3.665609 
3.556580 
3.621566 
– 
Std. dev. 
3.656607 
3.532003 
3.606272 
0.8223 
Log (var) 
3.656575 
3.531970 
3.606240 
0.8428 
It can be seen for the above that the information criteria do not improve for either model and likewise the pvalues for the repressor’s are insignificant, therefore we conclude there is no link between volatility and nominal exchange rate movements.
ii) asymmetric volatility.
To test for asymmetric volatility the above models will be tested again using EGARCH and TGARCH models, if they are significant and the information criteria improve we can conclude there is asymmetric volatility.
e_{t} AR(1) GARCH (1,1) 

AIC 
SC 
HQ 
Pvalue of regressor(s) 

GARCH 
3.647996 
3.554845 
3.610371 
– 
EGARCH 
3.597413 
3.504261 
3.559788 
0.0123, 0.0023, 0.0309 
TGARCH 
3.667624 
3.558948 
3.623728 
0.0005, 0.0000, 0.0000 
For the EGARCH model all of the regressors are significant however the information criteria are worse so we will discount this. For the TGARCH we can see that all of the information criteria are better and the regressors are all significant at the 5% significance level and therefore we can conclude there is evidence of asymmetric volatility and the TGARCH model should be used.
âˆ†e_{t} AR(1) MA(1) GARCH (1,1) 

AIC 
SC 
HQ 
Pvalue of regressor(s) 

GARCH 
3.665609 
3.556580 
3.621566 
– 
TGARCH 
3.710058 
3.585453 
3.659723 
0.0000, 0.0076, 0.0000 
EGARCH 
3.678623 
3.554019 
3.628288 
0.0381, 0.7862, 0.0024, 0.0014 
Similarly the results for the preferred model âˆ†e AR(1) MA(1) show evidence of asymmetric volatility and suggest that a TGARCH model should be used.
5. Given the results in above, reassess the forecasting performance of your competing models for et (12step ahead).
Forecast 

Model 
RMSE 
MAE 
MAPE 
U 
e AR(1) AR(4) 
0.0533860 
0.0476250 
1.7314010 
0.0098290 
e AR(1) 
0.0439990 
0.0368120 
1.3360190 
0.0080850 
âˆ†e AR(1) MA (1) 
0.0350707 
0.0276425 
1.0021864 
0.0064299 
âˆ†e constant 
0.03205544 
0.02585021 
0.93857607 
0.00586893 
âˆ†e AR(1) MA(1) TGARCH 
0.032527852 
0.026321482 
0.955692971 
0.005956007 
e AR(1) EGARCH 
0.038058768 
0.029901757 
1.083818142 
0.006983289 
As can be seen from the above results, despite adopting new models to account for the ARCH effects and asymmetric volatility the results still suggest that a constant model is the best forecast of the exchange rate. The models accounting for ARCH effects and asymmetric volatility did however perform better than there previous equivalents. It seems that using a constant estimator is the best forecast of exchange rates for the 12 month period however if looking further in to the future it would seem unlikely this would be the case as exchange rates fluctuate over time. If modelling further in to the future the model of choice would be the âˆ†e AR(1) MA(1) TGARCH.
6. Using the to your case, study (and test for) Grangercausality among exchange rates and prices.
The variables P_{t} (log(cpi_japan_sa)) and P_{ft} (log(cpi_sweden_sa)) were created in order to conduct the multivariate analysis and test the Purchasing Power Parity hypothesis (e_{t} = p_{t} – p_{ft} + u_{t}). Initially a VAR model with endogenous variables e_{t,} P_{t} and P_{ft} was estimated with an adjustedsample period to 2012m08 (to remove 12 periods for sampling). By inspecting the lag length criteria for 12 lags (due to the monthly data) it can be seen that 2 lags should be used as 3/5 information criteria suggest this with 1 other suggesting 1 lag and the other 10 lags.
Lag 
LogL 
LR 
FPE 
AIC 
SC 
HQ 
0 
1058.334 
NA 
3.78e09 
10.87973 
10.82919 
10.85927 
1 
2173.030 
2183.427 
4.24e14 
22.27866 
22.07653* 
22.19681 
2 
2188.028 
28.91262 
3.98e14* 
22.34049* 
21.98676 
22.19726* 
3 
2191.883 
7.312518 
4.20e14 
22.28745 
21.78211 
22.08283 
4 
2201.900 
18.69171 
4.16e14 
22.29794 
21.64100 
22.03192 
5 
2203.370 
2.697145 
4.50e14 
22.22031 
21.41176 
21.89290 
6 
2208.951 
10.06880 
4.66e14 
22.18506 
21.22492 
21.79627 
7 
2213.684 
8.392441 
4.88e14 
22.14107 
21.02932 
21.69089 
8 
2216.622 
5.118809 
5.20e14 
22.07857 
20.81523 
21.56701 
9 
2222.562 
10.16563 
5.37e14 
22.04703 
20.63208 
21.47408 
10 
2232.998 
17.53626* 
5.31e14 
22.06183 
20.49528 
21.42749 
11 
2238.369 
8.860814 
5.52e14 
22.02443 
20.30627 
21.32870 
12 
2245.094 
10.88415 
5.67e14 
22.00097 
20.13122 
21.24385 
The Granger Causality test was then conducted. From this it can be seen that at the 5% significance level we can reject the null of no causality from Pft to Pt and that we can reject the null of no causality from e_{t} and P_{ft} collectively to P_{t.} This is not a result you would expect to occur intuitively as it would not be expected that a change in P_{t} would be as a consequence of a change in P_{ft}.
Dependent variable: E 

Excluded 
Chisq 
df 
Prob. 
PT 
3.138309 
2 
0.2082 
PFT 
0.087914 
2 
0.957 
All 
4.648441 
4 
0.3253 
Dependent variable: PT 

Excluded 
Chisq 
df 
Prob. 
E 
5.63532 
2 
0.0597 
PFT 
13.94933 
2 
0.0009 
All 
15.13495 
4 
0.0044 
Dependent variable: PFT 

Excluded 
Chisq 
df 
Prob. 
E 
0.175002 
2 
0.9162 
PT 
2.819637 
2 
0.2442 
All 
4.540737 
4 
0.3377 
For the rest of the variables we fail to reject the null at the 5% significance level and therefore conclude there is no causality. This is not supporting of the PPP hypothesis in which you would expect to conclude causality from Pft and Pt to e_{t.}
7. Test whether or not et, pt and pft are stationary (several methods are possible, here).
The first step before testing for stationarity using the augmented DickeyFuller test is to inspect the graphs of the variables to establish if they have a trend and/or an intercept. It can be seen for the graphs of e_{t } and p_{t} that both have an intercept as they fluctuate around a nonzero mean, however neither appear to have a very obvious trend. Therefore when conducting the test only allowance for an intercept will be used but nothing for trend. From the graph of p_{ft} we can see that there is an intercept and that there is a clear upward trend therefore an allowance for intercept and for trend will be used. (see appendix ?? for graphs)
From conducting the ADF test and examining the pvalues we fail to reject the null hypothesis that the series has a unit root and therefore is nonstationary at the 5% significance level for all of the variables (et, pt and pft). In other words there is insufficient evidence to suggest that they are stationary.
Augmented DickeyFuller Test Results 

Variable 
Prob. 
e_{t} 
0.1411 
p_{t} 
0.6134 
p_{ft} 
0.7305 
âˆ†e_{t} 
0.0000 
âˆ†p_{t} 
0.0000 
âˆ†p_{ft} 
0.0000 
For each of the v
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