Econometrics and Purchasing Power Parity - Assignment Example

Econometrics 211 Assignment – Topic Purchasing Power Parity Introduction The purpose of this assignment is to test the theory underlying relative purchasing power parity (hereafter referred to as PPP). A regression model will be used to test the strength and direction of the relationships between foreign and domestic inflation rates with the change in the growth rate of the exchange rate between two countries – the US as the domestic country and Sri Lanka as the foreign country. The paper proceeds by giving a brief overview of some of the literature relevant to PPP. Thereafter the data and methodology used in the paper are discussed, before moving on to an examination of the regression output. A variety of diagnostics tests are then done on the regression in order to determine its validity. Finally, one of PPP’s underlying assumptions is tested by conducting t-tests on the coefficients of the regressors. Literature Review In its initial form, PPP holds that a bundle of goods in two countries should cost the same when inflation is taken into account. The major difference between this and relative PPP is that the latter incorporates the inflation rates in each country and suggests that these have a role to play in determining changes in the growth rate of the exchange rate between two countries. Relative PPP tends to focus on the long-run, and it holds that “over long periods of time exchange rates will tend to offset the differences in inflation rates between the two countries whose currencies comprise the exchange rate” (Simonoff 2006: 1). A brief survey of the literature shows that PPP is hardly a theory that is widely accepted in the academic world. In fact Simonoff (2006: 15) points out that for developing countries, the robustness of the principles of PPP is questionable. This is likely to be a problem in this assignment’s regression as the two countries in question – the US and Sri Lanka – are developed and developing respectively. This seemingly grey outlook is shared by many academics1 and Suranovic (1997: n.p.) goes so far as to say that “in general, the PPP theory holds miserably when applied to real world data.” This assignment will test whether this opinion is justified or not. The next section briefly introduces the data before embarking on the analysis. Data Description The data was sourced from the United Nations statistical data base and from the Penn World tables. The sample period runs from 1950 to 2006 and all data is annual. The fact that the data is annual means that seasonality is not a concern. The data includes the following variables: Name Description SLCPI Sri Lanka’s consumer price index with 2000 as the base year USCPI The United States’ consumer price index with 2000 as the base year ERATE The average annual exchange rate between the US and Sri Lanka d_ERATE The first difference of the exchange rates CHANGEX The first difference of the exchange rates divided by the exchange rate in order to give the growth rate of the exchange rate It was necessary to construct a variable capturing the growth rate of the exchange rate over time and so the CHANGEX variable was generated by simply dividing the first differenced exchange rate variable by the exchange rate variable. The final regression only makes use of the CHANGEX, SLCPI and USCPI variables. One of the features of the data is the fact that CPI2 for both countries displays an upward trend as to be expected. Sri Lanka’s CPI lies below that of the US until the late 1990s, whereupon it becomes significantly steeper and grows at a far faster rate than inflation in the US. This is shown on the graph on the following page. Regression Analysis: The output for the regression specified as is shown below. ∆x represents the change in the growth rate of the exchange rate, πtf is Sri Lanka’s inflation rate at time t and πt is the inflation rate in the US at time t and εt represents the random error terms. Model 1: OLS estimates using the 56 observations 1951-2006 Dependent variable: CHANGEX VARIABLE COEFFICIENT STDERROR T STAT P-VALUE const -0.00510837 0.0198960 -0.257 0.79836 SLCPI -0.00113524 0.000443616 -2.559 0.01339 USCPI 0.00187175 0.000601039 3.114 0.00297 Mean of dependent variable = 0.0504728 Standard deviation of dep. var. = 0.0698021 Sum of squared residuals = 0.225271 Standard error of residuals = 0.065195 Unadjusted R-squared = 0.159369 Adjusted R-squared = 0.127647 F-statistic (2, 53) = 5.02395 (p-value = 0.01) Durbin-Watson statistic = 2.0781 First-order autocorrelation coeff. = -0.0410885 A first glance at the regression output shows that there is a negative relationship between foreign inflation and the change in the growth rate of the exchange rate. This relationship is significant at the 5% level, as shown by the p-value. This is a somewhat strange result, given our previous expectations that the relationship would be negative. Likewise, the relationship between domestic inflation and the change in the growth rate of the exchange rate is surprising, given that it is positive and significant at the 5% level. The R-squared and adjusted R-squared values are rather low, with the former showing that about 16% of the change in the dependent variable can be explained by changes in the independent variables. This suggests that our model lacks something important in its current form. Crucially, however, the F statistic for 2 degrees of freedom in the numerator and 53 degrees of freedom in the denominator shows that the model is significant overall. This is because the F-statistic of 5.024 exceeds the critical F-value of 3.18. The fact that the constant is insignificant should not be a concern. The main finding is that the coefficients of the two variables are both significant at the 5% level. Distribution of the Residuals The graph above represents the distribution of the regression’s residuals. One of the assumptions of the classical linear regression model is that the residuals should be normally distributed. The null hypothesis is that the residuals are normally distributed, while the alternative hypothesis states that they are not normally distributed. The outlier for 1978 notwithstanding, the distribution appears, at first glance, to be normal. This is confirmed by the fact that once this outlier is ignored, the chi-squared calculated value falls below the critical, and we do not reject the hypothesis of normally distributed residuals. Multicollinearity The variance inflation factor for Sri Lankan inflation and US inflation is 5.43. Given the fact that these factors are less than 10, we can make the rule-of-thumb judgment and multicollinearity is not present in the model. This is further supported by the fact that another traditional sign of multicollineariy – high R-squared with low t-ratios – does not appear to be a property of this regression.3 Autocorrelation The graph below shows the residuals of the model plotted against time. If autocorrelation is present in the model, we could assume that the residuals would display some systematic patterns. Given that this is not a problem in the graph, we see that further tests are unlikely to reveal autocorrelation as a problem in this regression. Plotting the residuals against the lagged residuals, as shown below, also does not seem to indicate any systematic patterns. Finally, we note that the Durbin-Watson statistic is 2.0781. Assuming that n=60 and choosing a 5% significance level, the lower Durbin-Watson value is 1.51 while the upper value is 1.65. Given this, we do not reject the null hypothesis of no autocorrelation, and therefore conclude that autocorrelation is not a problem in this model. Heteroscedasticity Whites test for heteroskedasticity OLS estimates using the 56 observations 1951-2006 Dependent variable: uhat^2 VARIABLE COEFFICIENT STDERROR T STAT P-VALUE const -0.0297213 0.0211424 -1.406 0.16598 SLCPI -0.00297784 0.00571499 -0.521 0.60463 USCPI 0.00296471 0.00133939 2.213 0.03146 sq_SLCPI -1.32982E-05 1.27135E-05 -1.046 0.30060 SLCPI_USCPI 5.63316E-05 6.40593E-05 0.879 0.38341 sq_USCPI -3.95563E-05 1.92615E-05 -2.054 0.04526 ** Unadjusted R-squared = 0.103088 Test statistic: TR^2 = 5.772947, with p-value = P(Chi-square(5) > 5.772947) = 0.328943 The table above shows the results of White’s test for the presence of heteroscedasticity. Multiplying the number of observations by the R-squared of the new regression yields a value of 5.77. The critical Chi-squared value for this test with 2 degrees of freedom at a 5% level is 5.99. Because the calculated Chi-squared value is less than the critical value, we do not reject the null hypothesis of no heteroscedasticity and conclude that heteroscedasticity is not present in this regression. Functional Form Test Auxiliary regression for RESET specification test OLS estimates using the 56 observations 1951-2006 Dependent variable: CHANGEX VARIABLE COEFFICIENT STDERROR T STAT P-VALUE const 0.00137552 0.118491 0.012 0.99078 SLCPI 0.00124346 0.00702046 0.177 0.86012 USCPI -0.00224721 0.0116624 -0.193 0.84797 yhat^2 111.267 130.448 0.853 0.39767 yhat^3 -995.195 814.727 -1.222 0.22751 Test statistic: F = 4.472973, with p-value = P(F(2,51) > 4.47297) = 0.0162 The results of Ramsey’s RESET test are displayed in the table above. The critical value for this test at the 5% level of significance with 2 degrees of freedom in the numerator and 52 degrees of freedom in the denominator is 3.18. The computed value of 4.473 exceeds the critical value – therefore we conclude that the model is misspecified. This test, however, does not indicate what a more suitable functional form would be for this model. Possibly, the problem arises through using cross-sectional techniques on time-series data. Testing the Implicit Hypothesis that α1>0 and α2 Read More