# Multiple Regression and T- Test

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Multiple Linear Regression

A Brightwater car dealership, which serves the city of Brightwater and its surrounding communities, was taken over about four years ago by a group of investors led by Jake Rogers. Jake had previously studied marketing and economics at Brightwater University. After taking over the dealership, Jake decided to apply some of the knowledge he had gained from his studies to selling cars. After a few months of operation, he began experimenting with the price of cars and the monthly expenditure on radio advertising. He varied the price and advertising expenditure each month and kept track of the average rate of interest on automobile loans for the month. The data on per car price in thousands (Pr), advertising in thousands (Ad), interest rate (IR), month in which the values applied (Mth), and sales in the thousands (Sales) appear in the data table. Jake would like to know the functional relationship between sales, the price, advertising expenditure, and interest rate on car loans. He is also interested in determining whether there is a trend to the firm's sales.

a. See data is sheet 2 of Excel file with data for this assignment.

b. Run a multiple linear regression on the data file for part (2a). (EXCEL Data, Data, Analysis, Regression). Since "Sales" is the dependent (or regressor) variable, the sales data is the Input Y Range. The other 4 variables (Month, Price, Advertising, and Interest Rate) are independent (or explanatory) variables; the explanatory variable data is the Input X Range. (Hint: Excel will permit you to include multiple columns and rows in your X range.) (10 points)

c. From the regression results you obtain in part (2b), determine if each of the explanatory variables used in the regression is statistically significant at a 5 percent level (This means 2.5 percent in each tail of the distribution). You will need to use the t distribution table for this purpose. In your answer, make sure you state what the critical value of t is for each independent variable. The critical value is the value of t, such that if the t statistic for your independent variable (from your Excel output) is greater than the critical t or less than (-1) times the critical t, then you reject the null hypothesis. Hint: To determine the critical t, you will need both the level of significance (2.5% in this case) and the degrees of freedom. You will need to calculate the degrees of freedom, which is the number of observations minus the number of independent variables minus 1. Degrees of freedom = N-k-1. (10 points)

d. Rerun the regression using only those explanatory variables which were found to be significant in part (2c). Note that some of the explanatory variable coefficient estimates are positive and some are negative. Do the signs (positive or negative) on the explanatory variable coefficients make sense? Discuss. (10 points)

e. What purpose does the adjusted R2 serve? Discuss. (Note that we generally use adjusted R2 rather than R2 because it adjusts for the reduction of degrees of freedom as more explanatory variables are added into a regression model.) (5 points)

f. If, at current prices and interest rates, each additional sales dollar contributes $0.18 to profit before advertising and taxes, should the firm continue to advertise? Why or why not? Discuss. Hint: You will also need the relation between advertising and sales. You estimated this relation in the regression in part (2c). (10 points)

Second Exercise.

Below is the regression input and output (from Excel) for the regression in which the percentage change in the exchange rate is the dependent variable (Y) and the inflation differential (inflation rate home - inflation rate foreign) is the independent variable (X). According to the Purchasing Power Parity theory, the hypothesized coefficients (null hypothesis) in the regression are 0 (intercept coefficient) and 1 (inflation differential coefficient). The directions for Problem Set 1 instruct you to first read sections of the Madura textbook. You will need to do this to learn how to test if a coefficient is different from 1; it will explain how you calculate the t Statistic to test your null hypothesis from the Excel output. Excel will only directly provide a t statistic for the test of whether a coefficient is different from 0.

3a. Is there evidence to reject the null hypothesis that the intercept coefficient is equal to 0 at the 5% level of significance? (Hint: you will need a table with the t distribution to determine the critical t.) (10 points)

3b. Is there evidence to reject the null hypothesis that the inflation differential coefficient (Inf h - Inf f) is equal to 1 at the 5% level of significance? (Hint: you will need a table with the t distribution to determine the critical t. You will not use the t Statistic given in the Excel output, because it corresponds to the null hypothesis that the inflation differential is equal to 0.) (10 points)

3c. Overall, is there any evidence inconsistent with the Purchasing Power Parity theory? (Hint: The answer to this questions is yes, if you can reject at least one of the two above null hypotheses.) (10 points)

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#### Solution Summary

The solution provides explanations how to interpret the multiple regression output.

Hypothesis Testing : T-test of Multiple Regression Model and Critical Values

T test of a multiple regression model

Using observations on each variable, a computer program generated the following multiple regression model:

If the standard errors of the coefficients of the independent variables are, respectively,

, , , , and ,

can you conclude that the independent variable is needed in the regression model?

Let denote the coefficients of the variables in this model, and use a two-sided hypothesis test and significance level of to determine your answer.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the table.

The null hypothesis:

The alternative hypothesis:

The type of test statistic: t Degrees of freedom:

The value of the test statistic (Round to at least 2 decimal places):

The two critical values at the 0.05 level of significance (round to at least 2 decimal places):

Can you conclude that the independent variable x2 is needed in the regression model?

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