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Lecture 5_multiple regression_powerpoint2007

Lecture 5_multiple regression_powerpoint2007 - Lecture 5...

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Click to edit Master subtitle style 8/5/11 Lecture 5 Forecasting with Multiple
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8/5/11 Multiple Regression Model Formally, a multiple regression is a statistical procedure in which a dependent variable ( Yt ) is modeled as a function of multiple independent variables ( X1t, X2t, X3t, …, Xkt ). The multiple-regression model may be written as: where β0 is the intercept and the other β i ’s are the slope terms associated with the respective independent variables ( i.e ., the Xit’s ) with i = 1,2,…,k. t kt k t t t t X X X X Y ε β β β β β + + + + + + = 3 3 2 2 1 1 0 (5.1)
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8/5/11 Example 5.1 Following the example 4.3 presented in previous lecture, we believe that the US production index is not only influenced by the lagged short-term interest rate, but also follows a stochastic trend. This belief leads us to specify a model as: USPIt = β0 + β 1 USTBRt-1 + β 2 USPIt-1 + ε t (5.2) -A decline in the short-term interest rate will contribute to a decrease in the long-term interest rate, which in turn will stimulate the economy. We anticipate β 1 <0. - USPIt-1 is expected to be positive since the data signify a stochastic trend – with a rise in current production, the trend momentum will carry it to the next period, β 2 >0.
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8/5/11 Statistical Evaluation We obtain the estimated statistics presented in Table 5.1. Dependent Variable: USPI Method: Least Squares Sample: 1986M01 1999M07 Included observations: 163 Variable Coefficient Std. Error t-Statistic Prob. C 0.137173 0.358204 0.382947 0.7023 USTBR(-1) -0.073664 0.025625 -2.874719 0.0046 USPI(-1) 1.005558 0.003067 327.8179 0.0000 R-squared 0.998671 Mean dependent var 93.90620 Adjusted R-squared 0.998654 S.D. dependent var 12.25531 F-statistic 60117.97 Durbin-Watson stat 2.008455 Prob(F-statistic) 0.000000
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8/5/11 The Signs of Regression Coefficients The estimated equation is as follows: USPI t = 0.137 - 0.0737 USTBR t -1 + 1.0056 USPI t -1 , h =1,2,…,12 (5.3) -Are the estimated coefficients are consistent with our prior expectations? -What are the implications? -How reliable are our estimates?
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8/5/11 Test for Individual Significance Using t-statistic The t -statistic is designed to test the significance of an individual coefficient, such as H0: β 1 = 0 or β 2 = 0. To recall: - Since the t -distribution is symmetrical, we are concerned only with the absolute value of tcalc - If the calculated tcalc -statistic is greater than the critical level (using tcalc > 2 as an approximation), based on ( n -( k +1)) degrees of freedom ( df ), the null hypothesis of no correlation is rejected. - In our example, n = 163 and k = 2, so df =160. The tcalc - statistics are -2.87 for β 1 and 327.82 for β 2. Here both tcalc - statistics are as great deal above the critical value at the 5% level of significance, leading us to reject the null hypothesis of β 1 = 0 or β 2 = 0. / ) ˆ ( 1 1 β β - = calc t (standard error of ) 1 ˆ β (5.4)
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8/5/11 Test for Joint Significance Using the F-statistic Test the usefulness of the entire collection of predictor variables by using the F -test. The null hypothesis tested is that none of the Xt variables contributes to predicting Yt by setting the H0 : β 1 = β 2 …= β k = 0. If the null is rejected, then at least one variable contributes to the prediction of Yt . Thus, the alternative hypothesis is that at least one is nonzero.
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