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A Introduction and background information: This paper study the exchange rate function for the U.S. and England during the years 1957 to 1990 based on the monetary model and an autoregressive model. The monetary model is an important and influential model concerning two freely floating currencies. The exchange rate by definition is the relative price of moneys of the two corresponding countries. The major factors determining the exchange rate include the money supply, income, and interest rates of the countries. Formally let M be the money supply, GDP be the nominal GDP, IR be the nominal interest rate and X be the exchange rate. Furthermore, denote P as the price level, D as the real demand for money, E as the expected future rate of price inflation, r as the real interest rate. The model***[reference the green book] starts with the assumption of monetary equilibrium for both countries: (1) P = M/D, P* = M*/D*, where the superscript * indicates variables of the foreign country. The aggregate money demand function can be written as (2) D = KYIR^(-a), D* = K*Y*IR*^(-a), where K's are fixed constants and -a is the interest elasticity of demand for money. By I. Fisher, the nominal interest is the sum of real interest rate and the expected future inflation: (3) IR = r + E, IR* = r* + E*. By real interest rate parity we have (4) r = r*, implying that real rate of return on assets are the same in both countries. Lastly, assuming that the purchasing power parity holds, we have the following relationship: (5) P = XP*, showing how the price levels in the two countries are linked by the exchange rate. Using the exchange rate, one can convert the currency of one country to another. Equation (5) essentially states that the exchange rate equalizes the purchasing powers of the currencies at home and abroad, when expressed in a common currency unit. By substituting equations (1) to (4) into (5), one can reach the simple monetary model for exchange rate: (6) X = (M/M*)(K*/K)(Y*/Y)(IR/IR*)^a. To estimate the model, one can take transformation to attain the linear model: log X = the logarithmic log (M/M*) + log (K*/K) + log (Y*/Y) + a log (IR/IR*). There is much evidence in the literature that supports the claim that exchange rates follow approximately a random walk.***[reference the paper I xeroxed] If the variable X follows a random walk process, it can be written as X(t) = X(t-1) + epsilon(t) In other words, the exchange rate of the current period gives a good prediction for the exchange rate for the next period. The disturbance epsilon is introduced to reflect any relevant news and information that may affect the exchange rate. In the random walk model, the coefficient of X(t-1) is restricted to be one. Many events have effects that persist over time. To capture these effects, the model should include lagged variables. It is natural to extend the model to include other more previous values of X's in the model. This forms an autoregressive model of the exchange rate: X(t) = p1 X(t-1) + p2 X(t-2) + ... + pk X(t-k) + epsilon(t). Finally one can combine the above time series model with the basic monetary model, with additional lagged variables. In particular, we focus on the factors income and interest rate of the countries. Thus, the model studied in this paper is the following: M ER t 0 1i i 0 M M GDPUS t i IRUS t i 2i 5 j ER t j u t GDPEt i IRE t i i 0 i 0 Exchange rate is regressed on GDPR, which is the GDP ratio of the two countries; IRR, which is the interest rate ratio of the two countries; and the lag of exchange rate. The GDP ratio is obtained by dividing the GDPUS by GDPE. The interest rate ratio is obtained by dividing the IRUS by IRE. GDP measures the total market value of a country’s output and can be calculated by the expenditure approach using the equation GDP=C+I+G+(X-M). Interest rate, which is determined in the money market, affects investments. When interest rate increases, aggregate expenditure decreases because investment decreases. A decrease in aggregate expenditure lowers equilibrium output by a multiple of the initial decrease in investment. The purchasing power parity states that exchange rates are set so that the price of similar goods in different countries is the same. C. Report of empirical result The data set contains the variables ER, which is the exchange rate of U.S. dollars per English pound; GDPUS, which is the nominal U.S. GDP; GDPE, which is the nominal England GDP; IRUS, which is the 3 month Treasury bill rate for U.S.; IRE, which is the 3 month Treasury bill rate for England. The data were quarterly data from 1957 first quarter to 1990 third quarter. There are 135 observations in the data set. The statistics summary in Table 1 shows that the mean exchange rate is approximately two dollars per English pound from 1957 to 1990. The highest exchange rate occurred in the fourth quarter of 1957 where 2.8594 dollars were exchanged for an English pound. The lowest exchange rate occurred in the fourth quarter of 1985 where 1.1565 dollars were exchanged for an English pound. The mean GDP for U.S. is 12 times higher than the mean GDP for England. The mean interest rate for England is 1.3 times as much as the mean interest rate for U.S. The data for GDPs, which is not adjusted to inflation, is measured in nominal value. This supports the fact that the maximum and minimum of England’s GDP occurred in 1990 third quarter and 1957 first quarter respectively. The maximum and minimum of U.S. GDP occurred in 1990 third quarter and 1958 first quarter respectively. Table 1 Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------er | 135 2.235376 .5067401 1.1565 2.8594 gdpus | 135 1987.476 1518.92 441 5471.7 gdpe | 135 161.138 156.6626 21.67 549.26 irus | 135 6.104519 2.90844 1.02 15.09 ire | 135 8.305333 3.401131 3.18 16.04 gdpr | 135 15.88593 4.015345 9.915715 20.63329 irr | 135 .7432501 .1900515 .2073171 1.279551 The correlation matrix in Table 2 shows that multicollinearity is a problem in the data. The nearly unity correlation coefficient (0.9961) between GDPUS and GDPE shows that GDPUS and GDPE are highly intercorrelated. When multicollinearity is a problem, the OLS estimators, which are still BLUE, have large variances and covariances. The large variances make the estimate less precise. The tstatistics tend to be statistically insignificant but the R2 can still be very high. The confidence intervals are much wider and lead to the acceptance of the “zero null hypothesis” more readily. Also, the OLS estimators and their standard errors can be sensitive to small changes in the data if multicollinearity exists. One source of multicollinearity may be due to the data collection method employed, for example, sampling over a limited range of values taken by the regressors in the population. If the data collection method cannot be improved, we can combine cross-sectional and time-series data to alleviate the problem of multicollinearity given that the cross-sectional estimates do not vary substantially from one cross section to another. Table 2 er gdpr irr --------+-------------------------------------------------------------er | 1.0000 gdpr | -0.7932 1.0000 irr | -0.1066 -0.0105 1.0000 ***let’s add other variables including er1 er2 er3… and gdpr1 gpr2… In this study a linear model and a log model are considered. The objective is to choose a model that has homoskedastic and non-serial correlated error terms so that the OLS estimates are efficient. Based on the derivation of the monetary model, the log model is more appropriate that the linear model. Nevertheless, in this paper, the decision is made solely based on the statistical propoerty of the residualsheteroskedasticity and serial correlationto avoid GLS estimations which are more costly to perform. Based on empirical findings, the linear model is preferred because the log model exhibits heteroskedasticity. The White test and the Breush-Pagan test were used to test for heteroskedasticity. The 2 variance of the error term is assumed to be dependent on GDPR, IRR; the square of GDPR; the square of IRR; and the cross products of GDPR and IRR. The White test was inconclusive in determining whether the log model or the linear model should be used because heteroskedasticity is not observed in either model. For the White test of the regression of the squared of the residuals on the regressors, the null hypothesis that there is no heteroskedasticity cannot be rejected at the 1% significance level for both linear and log model. The Breusch-Pagan test, on the other hand, shows that the linear model is preferred because the log model exhibits heteroskedasticity. The Breusch-Pagan test runs the auxiliary regression of uˆi2 /( RSS / N ) on the regressors, where uˆi2 is the squared of the residuals and RSS is the residual sums of square and N is the number of observations. Using the Breusch-Pagan test, the null hypothesis that the log model has no heteroskedasticity is rejected at the 1% significance level. The t-statistic for the log model, which is 15.1004, is greater than the critical value of 15.0863 of the chi-square distribution with five degrees of freedom. For the test of heteroskedasticity of the linear model, the null hypothesis that the linear model has no heteroskedasticity is not rejected at the 1% significance level. The t-statistic of 6.808 for the linear model is smaller than the critical value of 15.0863. Refer to the table below for comparison. Linear Model (chi_sq_5, 0.01) Breusch -Pagan Test 6.808 WhiteTest 2.0541 Log Model Critical 15.1004 7.4169 Value 15.0863 15.0863 Decision rule: Reject null if t-statistic is greater than critical value of 15.0863 Heteroskedasiticity gives OLS estimators that are no longer BLUE. The OLS estimators are biased and they do not have the smallest variance among the class of unbiased least square estimators. If OLS estimation is used disregarding heteroskedasticity, the usual t and F tests are no longer valid. Applying those tests would give misleading conclusions about the statistical significance of the estimated regression coefficients. When heteroskedasticity exists, the GLS method of estimation should be used. GLS transforms the variables so that they satisfy the standard least-squares assumptions that variance of the disturbance term should equal a constant. LM tests for first order and fourth order autocorrelation are conducted but neither of the log models nor the linear model exhibits autocorrelation. The LM test for first order autocorrelation of the linear model regresses the residuals on GDPR, the eight lags of GDPR, IRR, the eight lags of IRR, the eight lags of ER, and one lag of the residuals. The null hypothesis that the linear model has no first order autocorrelation cannot be rejected at the 1% significance level because 4.575 is smaller than the critical value of 6.6349. The LM test for fourth order autocorrelation regresses the residuals on GDPR, the eight lags of GDPR, IRR, the eight lags of IRR, the eight lags of ER, and the four lags of the residuals. The test statistic of 8.3776 is smaller than the critical value of 13.2767 at the 1% significance level. Hence, the null hypothesis that the linear model has no fourth autocorrelation cannot be rejected. The LM tests cannot reject the null hypotheses that the model has no autocorrelation for the log model. For the LM test of first order autocorrelation, the log of the residuals are regressed on the log of GDPR, the log of the eight lags of GDPR, the log of IRR, the log of the eight lags of IRR, the log of the eight lags of ER, and the one lag of the log of the residuals. The null hypothesis that the log model has no first order autocorrelation cannot be rejected at the 1% significance level because 0.7686 is smaller than the critical value of 6.6349. The LM test of fourth order autocorrelation regressed the log of the residuals on the log of GDPR, the log of the eight lags of GDPR, the log of IRR, the log of the eight lags of IRR, the log of the eight lags of ER, and the four lags of the log of the residuals. The null hypothesis that the log model has no fourth order autocorrelation cannot be rejected at the 1% significance level because 4.3316 is smaller than the critical value of 13.2767. Refer to the table on the next page for comparison. Linear Model Log Model Critical Values st 1 order 4.575 0.7686 6.6349 (chi_sq, 1, 0.01) 4th order 8.3776 4.3116 13.2767 (chi_sq, 4, 0.01) Decision rule: Reject null if test statistic is greater than the critical value. If there exist autocorrelation but we disregard it in our OLS estimation, the residual variance( ˆ 2 ) is likely to underestimate the true 2 . As a result, R2 will tend to be overestimated. Therefore, the usual t and F tests of significance are no longer valid. Applying those tests would give misleading conclusions about the statistical significance of the estimated regression coefficients. Comparing the R2’s of the linear and log model, the results are inconclusive in determining whether to use the linear or the log model. First, ln ERˆ is obtained from the log model. Then, compute the exponential of lnER_hat. The squared correlation between exp(lnER_hat) and ER_hat is compared with R_square from the linear model. Squared correlation =0.9901^2=0.980298 between exp(lnER_hat) and ER_hat R_square from the linear model =0.9808 =>linear model with a higher R_square is preferred. Second, ER_hat is obtained from the linear model and the log of ER_hat is computed. Then the square of the correlation between ln(ER_hat) and lnER_hat is comapred with the R_square from the log model. Squared correlation between ln(ER_hat) and lnER_hat =0.9900^2=0.9801 R_squar e from the log model =0.9805 =>log model with a higher R_square is preferred. Based on the results from the Breusch-Pagan test that found that heteroskedasticity exists in the log model, the liner model was adopted in determining the lag length of the model. Different regressions were ran to determine how many lag lengths should be included in the model. The tstatistics are insignificant for any coefficients from the fifth to eight-order lags at the 1% significance level. The F-test of the coefficients are jointly zero are conducted and the null that the coefficients are jointly zeros are all accepted from the fifth to eighth-order lags. The fifth lags for ER, GDPR, and IRR are all dropped. The regression of ER on GDPR, the four lags of GDPR, IRR, the four lags of IRR, and the four lags of ER shows that the fourth lag of ER is significant at the 1% significance level. The test statistic of the fourth-lag is 0.008 is smaller than the 0.01 shows that the model should include the fourth-lag of ER. The F-test of the coefficients of the fourth-lag of GDPR and IRR are carried out and the null hypothesis that the coefficients are jointly zeros is accepted. A series of regressions and F-tests are carried out but the lag-length for GDPR and IRR are all insignificant at the 1% level from fourth-order to firstorder. Therefore, model 2 is determined to be: 4 GDPUS t i IRUS t i ER t 0 10 20 5 j ER t j u t GDPEt i IRE t i i 0 Model 2 is estimated and elasticities are calculated. the short-run and The short-run multiplier = 0.0095536 The short-run elasticity of ER with respect 0.0095536*(16.06696/2.251238) =0.06818 to long-run GDPR = The short-run elasticity of ER with respect to IRR = 0.0095536*(0.7497285/2.251238) =3.1816*10-3 The long-run multiplier = 0.0095536+(-0.064676)/[11.167024-(-0.3812324)-0.3507642 -(-0.2228933)] = -0.638452584 The long-run elasticity of ER with respect of GDPR=0.638452584*(16.06696/2.251238) =-4.5566 The long-run elasticity of ER with respect of IRR=0.638452584*(0.7497285/2.251238) =-0.2126 The predicted ERs for the last four periods are 1.639637 for 1990 first quarter, 1.565121 for 1990 second quarter, 1.673062 for 1990 third quarter, and 1.752046 for 1990 fourth quarter. The predicted accuracy test was carried out and the null hypothesis that the model is correctly specified is not rejected at Q equals 4, where Q is the number of periods for which predictions have been made. The test statistic for the Predictive Accuracy test is 4.238119, which is smaller than the critical value of 13.2767. D. Formal calculation of the hypothesis tests: Tests for heteroskedasticity of Linear Model: Breusch-Pagan Test Model: u_i_hat_sq/sigma_tilda_sq= alpha_1+alpha_2GDPR+alpha_3IRR +alpha_4GDPR_sq+alpha_5IRR_sq +alpha_6GDPR*IRR Null: sigma_i_sq=alpha_1=sigma_sq Alternative: sigma_i_sq= alpha_1+alpha_2GDPR+alpha_3IRR +alpha_4GDPR_sq+alpha_5IRR_sq +alpha_6GDPR*IRR Decision rule: Reject null if ESS/2 > chi_sq, 5, 0.01 ESS/2 =13.6163592/2 =6.808 chi_sq, 5, 0.01=15.0863 Since 6.808 < 15.0863, null hypothesis heteroskedasticity does not exist cannot be rejected. that White Test: Model: u_i_hat_sq =alpha_1+alpha_2GDPR+alpha_3IRR +alpha_4GDPR_sq+alpha_5IRR_sq +alpha_6GDPR*IRR Null: sigma_i_sq=alpha_1=sigma_sq Alternative: sigma_i_sq= alpha_1+alpha_2GDPR+alpha_3IRR +alpha_4GDPR_sq+alpha_5IRR_sq +alpha_6GDPR*IRR Decision rule: Reject null if NR_sq>chi_sq,5,0.01 NR_sq =123*0.0167 =2.0541 chi_sq, 5, 0.01=15.0863 Since 2.0541<15.0863, null hypothesis heteroskedasticity does not exist cannot be rejected. Tests for heteroskedasticity of Log Model: that Breusch-Pagan Test: Model: u_i_hat_sq/sigma_tilda_sq= alpha_1+alpha_2LGDPR+alpha_3LIRR +alpha_4LGDPR_sq+alpha_5LIRR_ sq +alpha_6LGDPR*LIRR Null: sigma_i_sq=alpha_1=sigma_sq Alternative: sigma_i_sq= alpha_1+alpha_2LGDPR+alpha_3LIRR +alpha_4LGDPR_sq+alpha_5LIRR_sq +alpha_6LGDPR*LIRR Decision rule: Reject null if ESS/2 > chi_sq, 5, 0.01 ESS/2 =30.200814/2 =15.1004 chi_sq, 5, 0.01=15.0863 Since 15.1004>15.0863, null hypothesis heteroskedasticity does not exist is rejected. that White Test: Model: u_i_hat_sq =alpha_1+alpha_2LGDPR+alpha_3LIRR +alpha_4LGDPR_sq+alpha_5LIRR_ sq +alpha_6LGDPR*LIRR Null: sigma_i_sq=alpha_1=sigma_sq Alternative: sigma_i_sq= alpha_1+alpha_2LGDPR+alpha_3LIRR +alpha_4LGDPR_sq+alpha_5LIRR_sq +alpha_6LGDPR*LIRR Decision rule: Reject null if NR_sq>chi_sq,5,0.01 NR_sq =123*0.0603 =7.4169 chi_sq, 5, 0.01=15.0863 Since 7.4169<15.0863, null hypothesis heteroskedasticity does not exist cannot be rejected. that LM Tests for autocorrelation for the linear model: LM test for 1st order autocorrelation: Null: Alternative: rho=0 not(rho=0) Decision rule: Reject null if (T-1)R_sq>chi_sq, 1, 0.01 (T-1)*R_sq =122*0.0375 =4.575 chi_sq, 1, 0,01=6.6349 Since 4.575 < 6.6349, the null hypothesis that there is no autocorrelation is not rejected. LM test for 4th order autocorrelation: Null: Alternative: rho_1=rho_2=rho_3=rho_4=0 not (rho_1=rho_2=rho_3=rho_4=0) Decision rule: Reject null if (T-4)R_sq>chi_sq, 4, 0.01 (T-4)*R_sq =119*0.0704 =8.3776 chi_sq, 1, 0,01=13.2767 Since 8.3776 < 13.2767, the null hypothesis that there is no autocorrelation is not rejected. LM Tests for autocorrelation for the log model: LM test for 1st order autocorrelation: Null: Alternative: rho=0 not(rho=0) Decision rule: Reject null if (T-1)R_sq>chi_sq, 1, 0.01 (T-1)*R_sq =122*0.0063 =0.7686 chi_sq, 1, 0,01=6.6349 Since 0.7686 < 6.6349, the null hypothesis that there is no autocorrelation is not rejected. LM test for 4th order autocorrelation: Null: Alternative: rho_1=rho_2=rho_3=rho_4=0 not (rho_1=rho_2=rho_3=rho_4=0) Decision rule: Reject null if (T-4)R_sq>chi_sq, 4, 0.01 (T-4)*R_sq =119*0.0364 =4.3316 chi_sq, 1, 0,01=13.2767 Since 4.3316 < 13.2767, the null hypothesis that there is no autocorrelation is not rejected. Predictive Accuracy Test of Model 2: Define: u T+j|T_hat = Y T+j|T – Y T+j|T_hat PA = summation chi_sq_Q j=1 to Q uT+j|T_hat_sq / sigma_hat_sq ~ Sigma_hat_sq is the estimated variance for the disturbance term for Model 2 and Q is the number of periods for which predictions have been made. Null: model is correctly specified Alternative: model is misspecified Decision rule: Reject null f PA > chi_sq, 4, 0.01 PA =0.0011654+0.006034+0.0047249+0.0147512 / 0.00294065 =4.238119 chi_sq,4, 0.01=13.2767 Since 4.238119 < 13.2767, the null hypothesis model is correctly specified is not rejected. that the