Hypothesis Testing in the Linear Regression Model An Overview of t tests, D Prescott

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1 Hypothesis Testing in the Linear Regression Model An Overview of t tests, D Prescott 1. Hypotheses as restrictions An hypothesis typically places restrictions on population regression coefficients. Consider the following model of house prices: Here: S = house size, LS = lot size, F = 1 if fireplace else 0, BR = 1 if busy road else 0, T = date of house sale (months from January 1983). Example 1: The hypothesis that the presence of a fireplace has no effect on the price of houses implies the coefficient on F is zero: : 0 1 This hypothesis restricts the coefficient to be zero. Example 2: The hypothesis that being on a busy road reduces house prices by $4000 is written as: : Example 3: The hypothesis that an extra square foot of living space adds the same value to house prices as an extra 50 square feet of lot is written: : 50 3 It is convenient to express this hypothesis in equivalent terms: namely that some new parameter equals zero, since testing parameters are zero is straightforward. 50 Hypothesis [3] can be written as: : 0 3 We need to introduce the new parameter into the regression model by eliminating. The regression model can be written as: 50

2 Another rearrangement gives the following equivalent regression model: 50 To estimate this model we create a new variable Z = LS + 50S and then estimate the following model 2. t statistics Standardized random variables are created by subtracting their expected value and dividing by their standard deviation or estimated standard deviation (referred to as a standard error). This applies to regression coefficients. For example the least squares coefficient on F is b 4. Its mean over many samples is. The standard error of b 4 is calculated by the regression program we can refer to it as SE(b 4 ) The standardized version of b 4 is distributed as the t distribution with a variance slightly larger than 1, but very close to 1 as the sample size tends to gets large. 5 The test statistic we use to test an hypothesis about we substitute the hypothesised value into the t statistic. The test statistics for the hypotheses [1], [2] and [3] are these:

3 3. Critical Values When the null hypothesis it true, the standardized coefficient or t statistic has a mean of zero. If the t statistic is far away from zero it implies the sample information (embodied in the least squares coefficient) is very different from the hypothesised value. See the numerator of the t statistics. The numerator of the t statistic measures the difference between what the data says the coefficient is and what the hypothesis says. If this difference is big enough we reject the hypothesis we decide the hypothesis is false. Given our large sample sizes the critical values are given by the normal distribution (which is very close to the relevant t distribution.) For a test at the 5% level of significance, the critical values are +/ 1.96 The hypothesis is rejected if the t statistic > Examples continued The examples use the TSP program. Students using R can ignore the programming details in TSP and focus on the structure of the t tests. In TSP the cdf function calculates critical values. Note that for a t test with 2489 = n 5 degrees of freedom the critical value for a test at the 5% level of significance is 1.96 Hypothesis [1] The t statistic for the hypothesis the true coefficient is zero is the ratio of the coefficient to its standard error. This is automatically computed by TSP and reported in the 4 th column of the regression output. Hence, TSP s reported t statistics (4 th column) are relevant to the hypotheses that the population coefficients are each zero. For hypothesis [1], the computed t statistic is 17.9 which is much greater than The hypothesis is rejected. Hypothesis [2] The t statistic is computed in the TSP program and printed the value is Since this is less than 1.96, the hypothesis that being on a busy road reduces process by $4000 is not rejected. The point estimate of $1754 is quite different from the hypothesised value of $4000 but the standard error is so large, the point estimate is not particularly reliable. The evidence against the hypothesis is too weak for it to be rejected. Hypothesis [3] The test statistic is the coefficient on size in the second regression of the TSP output. Notice that this coefficient is the difference between (in the first regression) the coefficient on size minus 50 times the coefficient on lot size about 0.2. The t statistic is 0.04 which is less than 1.96, so the hypothesis is not rejected. Note that 50 times the coefficient on lot size is 50*0.942 = This is very close to the coefficient on size, which is The data do not conflict greatly with the null hypothesis, so the hypothesis is not rejected.

4 COMMAND *************************************************************** 1 read(file='mls_1.txt',format='(3(f8.0,1x),3(f4.0,1x),8(f3.0,1x))') 1 price size lsize age bathp T poolag poolig 1 fp sgar dgar car_p busy_rd ele ; 2 2 cdf(inv,t,df=2489) 0.05; 3 3 cdf(inv,t,df=2489) 0.01; 4 4 ols price c size lsize FP busy_rd T; 5 5? 5? Hypothesis [2] 5? 5 5 set t2 = (@coef(5)+4000)/@ses(5); 6 t2; 7 7? 7? Hypothesis [3] 7? 7 7 Z = lsize + 50*Size; 8 8 ols price c size Z FP busy_rd T; ******************************************************************** Current sample: 1 to 2494 T(2494) Critical Value: , Two-tailed area: T(2494) Critical Value: , Two-tailed area: Equation 1 ============ Method of estimation = Ordinary Least Squares Dependent variable: PRICE Number of observations: 2494 Mean of dep. var. = Std. dev. of dep. var. = Sum of squared residuals = E+12 Variance of residuals = E+09 Std. error of regression = R-squared = Estimated Standard Variable Coefficient Error t-statistic P-value C [.000] SIZE [.000] LSIZE [.000] FP [.000] BUSY_RD [.150] T T2 Value

5 Equation 2 ============ Method of estimation = Ordinary Least Squares Dependent variable: PRICE Number of observations: 2494 Mean of dep. var. = Std. dev. of dep. var. = Sum of squared residuals = E+12 Variance of residuals = E+09 Std. error of regression = R-squared = Estimated Standard Variable Coefficient Error t-statistic P-value C [.000] SIZE [.970] Z [.000] FP [.000] BUSY_RD [.150] T [.000]