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Unformatted text preview: Click to edit Master subtitle style 8/5/11 Lecture 5 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 multipleregression model may be written as: where 0 is the intercept and the other i s are the slope terms associated with the respective independent variables ( i.e ., the Xits ) with i = 1,2,,k. t kt k t t t t X X X X Y + + + + + + = 3 3 2 2 1 1 (5.1) 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 shortterm interest rate, but also follows a stochastic trend. This belief leads us to specify a model as: USPIt = 0 + 1 USTBRt1 + 2 USPIt1 + t (5.2)A decline in the shortterm interest rate will contribute to a decrease in the longterm interest rate, which in turn will stimulate the economy. We anticipate 1 <0. USPIt1 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, 2 >0. 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 tStatistic Prob. C 0.137173 0.358204 0.382947 0.7023 USTBR(1)0.073664 0.0256252.874719 0.0046 USPI(1) 1.005558 0.003067 327.8179 0.0000 Rsquared 0.998671 Mean dependent var 93.90620 Adjusted Rsquared 0.998654 S.D. dependent var 12.25531 Fstatistic 60117.97 DurbinWatson stat 2.008455 Prob(Fstatistic) 0.000000 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? 8/5/11 Test for Individual Significance Using tstatistic The tstatistic is designed to test the significance of an individual coefficient, such as H0: 1 = 0 or 2 = 0. To recall: Since the tdistribution is symmetrical, we are concerned only with the absolute value of tcalc If the calculated tcalcstatistic 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 1 and 327.82 for 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 1 = 0 or 2 = 0....
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 Spring '09

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