Multiple Regression Model - Statistics Project Example

MULTIPLE REGRESSION MODEL Introduction In this paper, we present an econometric model for forecasting a dependent variable. The data are derived from the World Bank database. The dependent variable of interest is the gross domestic product (GDP) growth rate. Tow explanatory variables have been employed to help explain the dependent (response) variable. We evaluate the appropriateness of the forecast model as well in the subsequent sections. Description of the dependent variable In this section we describe the dependent (response) variable used in our econometric model. We have used GDP growth rate as the dependent variable. The variable is chosen because it is one of the most important indicators of economic health. For instance, when the economy is expanding, we expect the GDP growth rate to be positive. On the other hand, if the GDP growth rate is actually turning negative, then the countrys economy is heading towards or is already in a recession. The annual percentage growth rate of the GDP at market prices is based on constant local currency. The aggregates are based on constant 2005 U.S. dollars. It is worth noting that GDP is the sum of gross value added by all resident producers in the economy plus any product taxes and minus any subsidies not included in the value of the products. It is calculated without making deductions for depreciation of fabricated assets or for depletion and degradation of natural resources. The data used was collected from the World Bank database. It is an annual data set spanning from 1970 to 2013 (with 44 observations). The figure below presents the time series plot for the variable. Figure 1: Time series plot for GDP growth rate From figure 1 above, we observe that the highest GDP growth rate for the UK was recorded in early 1970s, with a high of 7.24%. On the other hand, the lowest ever recorded GDP growth rate was the period around 2008-2009 (during the global financial crisis). This is the period UK entered a recession hence explaining the low growth rate recorded during this period. Econometric model Econometric models are statistical models used in econometrics. An econometric model specifies and describes the statistical relationship that is believed to exist between the various economic quantities pertaining to a particular economic phenomenon under study. An econometric model can be derived from a deterministic economic model by allowing for uncertainty, or from an economic model which itself is stochastic. However, it is also possible to use econometric models that are not tied to any specific economic theory (Sims, 1980). In this section, we describe the econometric model used. Multiple linear regression is the model used. The model is used to study the relationship between a dependent variable and one or more independent variables. The generic form of the linear regression model is In our case, we intend to have two independent variables that would explain the variation in the dependent variable (GDP growth rate). The two independent (explanatory) variables are inflation rate in UK and the UK unemployment rate. The model is thus supposed to be as shown below; Where, represents the coefficient for the intercept represents the coefficient for the parameter inflation rate represents the coefficient for the parameter unemployment rate Inflation is a condition, when cost of goods and services rise and the entire economy seems to blink. Inflation has never done any good to the economy and as such we expect inflation rate to have a negative relationship with the GDP growth rate. Inflation and economic growth (GDP growth rate) are always in parallel lines and can never meet. Inflation reduces the value of money and makes it difficult for the common people. It is always bad news when the unemployment rate is high. High unemployment is a matter of concern for everyone and as such it affects the economic growth of a country. We expect the GDP growth rate to go down with high unemployment rate. Estimation of the model In this section we present the estimated model based on the dependent (response) and independent (explanatory) variables mentioned above using Eviews software. Using EViews, we estimate the initial version of the model, but dropping the last 5 years from the data set (that is, the years 2009 to 2013). The results are shown in table 1 below. Table 1: Initial model Dependent Variable: GDP_GROWTH__ANNUAL___ Method: Least Squares Date: 01/04/15 Time: 09:16 Sample (adjusted): 1970 2008 Included observations: 39 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   C 4.060901 0.861904 4.711549 0.0000 INFLATION_RATE -19.77709 5.594755 -3.534934 0.0011 UNEMPLOYMENT_RATE -1.794489 7.943823 -0.225897 0.8226 R-squared 0.269739     Mean dependent var 2.580881 Adjusted R-squared 0.229169     S.D. dependent var 2.006371 S.E. of regression 1.761533     Akaike info criterion 4.044049 Sum squared resid 111.7079     Schwarz criterion 4.172016 Log likelihood -75.85896     Hannan-Quinn criter. 4.089962 F-statistic 6.648727     Durbin-Watson stat 1.528189 Prob(F-statistic) 0.003488 Table 1 above gives the initial model (with 5 observations dropped). We see that only 26.97% of the variation in the dependent variable (GDP growth rate) is explained by the independent variables in the model. However, the p-value for the F-Statistic is very significant (with a value of 0.003) at 5% significance level, meaning that the model is significant and fit. Diagnostic tests To check on the appropriateness of the model, we did some few diagnostic tests. One of the tests was Breusch-Pagan-Godfrey test for heteroscedasticity. The test is used to test whether the estimated variance of the residuals from a regression are dependent on the values of the independent variables. In such a case, we have heteroskedasticity in our model. The results are presented in table 2 below. Table 2: Test for heteroscedasticity Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic 0.689768     Prob. F(2,36) 0.5082 Obs*R-squared 1.439342     Prob. Chi-Square(2) 0.4869 Scaled explained SS 2.018334     Prob. Chi-Square(2) 0.3645 From the table, we observe the p-value to be 0.5082 (a value greater than 5% significance level) we thus fail to reject the null hypothesis and conclude that there is constant variance (presence of homoscedasticity) in the data. This shows that the data does not need any kind of adjustment or transformation. Apart from the test for heteroscedasticity, we also conducted test for serial autocorrelation. We used Breusch–Godfrey-Bertolo test to assess the validity of the modelling assumptions inherent in a regression-like models. This test examines for the presence of serial dependence that may not have been included in a proposed model structure and which, if present, would mean that incorrect conclusions would be drawn from other tests. The table below gives the results; Table 3: Test for serial autocorrelation Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.827679     Prob. F(2,34) 0.4457 Obs*R-squared 1.810638     Prob. Chi-Square(2) 0.4044 The null hypothesis is that there is no serial correlation. In the table, we see the p-value to be 0.4457 (a value greater than 5% significance level), we thus fail to reject the null hypothesis and conclude that there is no serial correlation in the model. Re-estimation of the model In this section, we present the final model. This model includes all the variables from 1970-2013. The model has been selected based on the fact that the initial model was found to be fit and that it had taken into consideration the key assumptions associated with the OLS. Table 4: Re-estimated OLS model Dependent Variable: GDP_GROWTH__ANNUAL___ Method: Least Squares Date: 01/04/15 Time: 13:31 Sample: 1970 2013 Included observations: 44 Variable Coefficient Std. Error t-Statistic Prob.   C 2.773264 1.029960 2.692594 0.0102 INFLATION_RATE -11.75154 6.722101 -1.748195 0.0879 UNEMPLOYMENT_RATE 3.286144 9.870358 0.332931 0.7409 R-squared 0.385071     Mean dependent var 2.279144 Adjusted R-squared 0.340440     S.D. dependent var 2.251834 S.E. of regression 2.205831     Akaike info criterion 4.485832 Sum squared resid 199.4933     Schwarz criterion 4.607481 Log likelihood -95.68830     Hannan-Quinn criter. 4.530945 F-statistic 11.90614     Durbin-Watson stat 1.322115 Prob(F-statistic) 0.001600 Using table 4 above we estimate the model as follows; Generating forecasts over 5 years We generated the 5 year forecast values for the dependent variable (GDP growth rate). Looking at the forecast values, we observe that the values are closely related to the actual values; this shows that the forecasting performance does not suggest any further improvements that could be made to your model. For 95% confidence intervals, t(41, 0.025) = 2.020 Obs GDP_growth__ann prediction std. error 95% interval 2009 -5.17041 3.04562 2.28811 (-1.57532, 7.66656) 2010 1.65975 2.49230 2.23371 (-2.01876, 7.00336) 2011 1.11738 2.41850 2.23206 (-2.08924, 6.92624) 2012 0.277723 2.66668 2.24008 (-1.85726, 7.19062) 2013 1.74351 2.67704 2.24197 (-1.85072, 7.20479) Actual values Year GDP growth (annual %) 2013 1.74351 2012 0.277723 2011 1.11738 2010 1.65975 2009 -5.17041 Critical evaluation of the econometric approach The table below gives the forecast evaluation statistics. The following evaluation statistics are given the mean error (ME), the mean squared error (MSE), the mean absolute error (MAE), the mean percentage error (MPE), the mean absolute percentage error (MAPE), and Theils U-statistics. The lower the value of the U1 statistic, the more accurate the forecasts are. The U1 statistic is bounded between 0 and 1, with values closer to 0 indicating greater forecasting accuracy. The table gives the value of U1 to be 0.24, a value closer to zero thus showing a more accurate forecast. Thus the forecasts made are appropriate. Also, a low value of the Mean Error (ME) may conceal forecasting inaccuracy due to the offsetting effect of large positive and negative forecast errors. In the table, we observe the value of Mean Error (ME) to be 2.7344 (a value greater than zero), this further confirms the accuracy of the forecast model. Forecast evaluation statistics Mean Error 2.7344 Mean Squared Error 15.294 Root Mean Squared Error 3.9107 Mean Absolute Error 2.7344 Mean Percentage Error -184.29 Mean Absolute Percentage Error 184.292 Theils U 0.23853 Bias proportion, UM 0.48891 Regression proportion, UR 0.41873 Disturbance proportion, UD 0.092359 References Sims, Christopher A. (1980). Macroeconomics and Reality. Econometrica 48 (1): 1–48 Hughes Hallett, Andrew J. Econometrics and the Theory of Economic Policy: The Tinbergen-Theil Contributions 40 Years On,"Oxford Economic Papers (1989) 41#1 pp 189–214 APPENDIXES Year Unemployment rate GDP growth (annual %) Inflation rate 2013 7.80% 1.74351 3.00% 2012 8.20% 0.277723 3.20% 2011 7.80% 1.11738 5.20% 2010 7.90% 1.65975 4.60% 2009 6.50% -5.17041 -0.50% 2008 5.20% -0.769484 4.00% 2007 5.50% 3.42724 4.30% 2006 4.70% 2.755 3.20% 2005 4.70% 3.2348 2.80% 2004 4.80% 3.17315 3.00% 2003 5.00% 3.94892 2.90% 2002 5.10% 2.2952 1.70% 2001 5.30% 2.18489 1.80% 2000 5.90% 4.36227 3.00% 1999 5.80% 2.93836 1.50% 1998 6.30% 3.56703 3.40% 1997 7.40% 4.35043 3.10% 1996 9.80% 3.49131 2.40% 1995 10.80% 3.53339 3.50% 1994 12.20% 4.95341 2.40% 1993 13.40% 3.49049 1.60% 1992 12.70% 1.29473 3.70% 1991 10.60% -1.29244 5.90% 1990 7.70% 0.779266 9.50% 1989 8.30% 2.28145 7.80% 1988 10.70% 5.03196 4.90% 1987 13.30% 4.56222 4.20% 1986 14.80% 4.01233 3.40% 1985 14.50% 3.59938 6.10% 1984 14.10% 2.67092 5.00% 1983 13.00% 3.62435 4.60% 1982 11.90% 2.09307 8.60% 1981 10.20% -1.32361 11.90% 1980 6.70% -2.10281 18.00% 1979 5.20% 2.60765 13.40% 1978 5.60% 3.34461 8.30% 1977 5.70% 2.29095 15.80% 1976 5.60% 2.7716 16.50% 1975 4.10% -0.556251 24.20% 1974 2.60% -1.58453 16.00% 1973 2.60% 7.24487 9.20% 1972 3.70% 3.61866 7.10% 1971 3.40% 2.08142 9.40% 1970 2.60% 2.66817 6.40% Test for autocorrelation Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.827679     Prob. F(2,34) 0.4457 Obs*R-squared 1.810638     Prob. Chi-Square(2) 0.4044 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 01/04/15 Time: 09:46 Sample: 1970 2008 Included observations: 39 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob.   C -0.154922 0.874980 -0.177057 0.8605 INFLATION_RATE 0.990436 6.063703 0.163338 0.8712 UNEMPLOYMENT_RATE 1.006601 8.261287 0.121846 0.9037 RESID(-1) 0.222202 0.188273 1.180210 0.2461 RESID(-2) -0.134818 0.205509 -0.656019 0.5162 R-squared 0.046427     Mean dependent var -1.94E-16 Adjusted R-squared -0.065758     S.D. dependent var 1.714550 S.E. of regression 1.770026     Akaike info criterion 4.099074 Sum squared resid 106.5217     Schwarz criterion 4.312352 Log likelihood -74.93195     Hannan-Quinn criter. 4.175596 F-statistic 0.413839     Durbin-Watson stat 1.862601 Prob(F-statistic) 0.797428 Test for heteroscedasticity Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic 0.689768     Prob. F(2,36) 0.5082 Obs*R-squared 1.439342     Prob. Chi-Square(2) 0.4869 Scaled explained SS 2.018334     Prob. Chi-Square(2) 0.3645 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 01/04/15 Time: 09:45 Sample: 1970 2008 Included observations: 39 Variable Coefficient Std. Error t-Statistic Prob.   C 3.143605 2.597134 1.210413 0.2340 INFLATION_RATE 12.81638 16.85841 0.760237 0.4521 UNEMPLOYMENT_RATE -14.85638 23.93675 -0.620652 0.5387 R-squared 0.036906     Mean dependent var 2.864306 Adjusted R-squared -0.016599     S.D. dependent var 5.264432 S.E. of regression 5.307945     Akaike info criterion 6.250090 Sum squared resid 1014.274     Schwarz criterion 6.378056 Log likelihood -118.8768     Hannan-Quinn criter. 6.296003 F-statistic 0.689768     Durbin-Watson stat 1.815046 Prob(F-statistic) 0.508200 Read More