Regression Analysis Homework 1 This homework is due Wednesday, September 14th, in class BEFORE class starts. Late papers will not be accepted. Please remember to staple if you turn in more than one page. You must SHOW ALL WORK. If you do not show your work, you may not receive full credit. 1 a In the simple regression model, suppose each Xi is replaced by cXi , where c 6= 0 is a constant. How are the parameters, b0 , b1 , b, the coefficient of determination, R2 , and the t-test of the null hypothesis H0 : 1 = 0 affected? b Suppose each Yi is replaced by dYi , for d 6= 0. Repeat part (a). c What do you conclude based on the results in part (a) and (b)? 0 0 2 Suppose p both X and Y are transformed by X = (X X)/sX and Y = (Y Y )/sY , where sx = SXX /(n 1) (the sample deviation of X1 , . . . , Xn ) and sY is similarly defined. b0 and R02 be the parameter estimators and the coefficient of determinations Let b00 , b10 , calculated from the transformed variables. i Compute b00 , b10 , b0 and R02 and compare them to b0 , b1 , b and R2 . What well known quantity is b10 ? ii Is there any change in the t-test of the null hypothesis 1 = 0 from the untransformed variables to the transformed ones? 3 Consider the following simple linear regression model: y = 0 + 1 x + \u000f (1) and assume the parameter 0 is known. a Find the least squares estimator for 1 . b Compute V ar(b10 ,known ). c Is V ar(b1 0 ,known ) V ar(b1 0 ,unknown )? Provide a formal proof. 4 Sampling Distributions for Sample Mean and Sample Variance Using R Suppose that an experimenter observes a set of variables that are taken to be normally distributed with an unknown mean and variance. Using simulation methods, for given values of the mean and variance, we can simulate the data values that the experimenter might obtain. More interestingly, we can simulate lots of possible samples of which, in reality, the experimenter would observe only one. Performing this simulation allows us to check on sampling distributions of the parameter estimates. Let us assume that = 100 and 2 = 9, which, in fact, the experimenter does not know. In our simulation study, we assume that the experimenter will observe 100 observations, which are normally distributed. To simulate a sample of 100 observations from N (100, 9), which the experimenter might observe, the R command is 1 Question 4. What is the (theoretical) sampling distribution of b if we know that the 500 samples come from a normal distribution N (100, 9)? Does the histogram approximate the sampling distribution for the sample mean? Why? Question 5. What is the (theoretical) sampling distribution of b2 if we know that the 500 samples come from a normal distribution N (100, 9)? Does the distribution of the sample variances from the 500 samples approximate the theoretical sampling distribution? Instructions. In order to evaluate the sampling distribution for b2 = S 2 , we can use the qqplot. We will use qqplot to compare the distribution of the sample for b2 and its theo2 retical distribution. The function used for the sampling distribution of b is rchisq(500, df = (100-1)) and the R command for using qqplot function is qqplot(variances, (rchisq(500, df = (100-1))*9/(n-1))) Please learn how to use the R command qqplot to compare the theoretical distribution with the approximated distribution based on the 500 samples. Question 6. Run the same simulation but for a smaller number of samples, say S = 50, and compare the sample mean and sample variance distributions based on this smaller number of samples to the sampling distributions for mean and variance based on the 500 samples and to the theoretical sampling distributions. 3