Question

In a study for housing demand, the following regression model was estimated. The standard errors of each coefficient are shown in the parentheses below. log Qt = 4.17 – 0.24 log Pt + 0.96 log Yt + 0.46 log MOR15t - 0.52 MOR30t + εt (0.03) (0.32) (0.23) (0.40) Adj R 2= 0.84, DW = 2.75, N=30. Where, Q = quantity of housing demanded P = price of unit of housing Y = family income MOR15 = 15-year mortgage rate MOR30 = 30-year mortgage rate

A. Do the “signs” of coefficient match your prior expectation? Examine each coefficient and explain why or why not.

B. Find the critical t-value and test the statistical significance of individual coefficients (for P, Y, MOR15 and MOR30) with 95% confidence level. Show your calculations.

C. What is the nature of “autocorrelation” problem in regression? Explain. Test whether the autocorrelation is a problem in the estimation above. Show your work.

D. What is the multicollinearity problem in regression analysis? Explain. Do you see any potential multicollinearity problem in the model above? Why or why not?

Answer 1

A. When we look at price of unit of housing we expect the sign to be negative as price and quantity of a good are inversely related. The results also agree with our expectation as as quantity demanded of houses decreases by 0.24% when we increase price by 1%.

When we look at family income we expect the sign to be positive as higher the income higher the quantity of houses demanded. The results also agree with our expectation as as quantity demanded of houses increases by 0.96% when we increase family income by 1%.

When we look at 15-yr mortgage rate we expect the sign to be negative as mortgage rate is nothing but loan and when we increase our mortgage by 1% we expect quantity of houses demanded to fall. The results dont agree with our expectation as as quantity demanded of houses increases by 0.46% when we increase mortgage by 1%.

Same, When we look at 30-yr mortgage rate we expect the sign to be negative as mortgage rate is nothing but loan and when we increase our mortgage by 1% we expect quantity of houses demanded to fall. The results agree with our expectation as as quantity demanded of houses decreases by 52% when we increase mortgage by 1%.

B. t- ratio

t -ratio = _j / std. er(_j), j = explanatory variable , here j = 1,2,3,4

H0 : j = 0

Ha : j is not equal to zero

We calculate t-ratio and then compare the t-calculated with t-critical at % level of significance. If t-calculated is greater than t-critical at % level of significance we reject null hypothesis and state that particular variable is statistically significant. Also if our t-ratio is significantly greater than 2.00, we term that variable is statistically significant.

T-ratio for price of houses = 1 / std er(1)

= - 0.24/0.03 = - 8

T-ratio for family income =  2/ std er(2)

= 0.96/0.32 = 3

T-ratio for 15-yr mortgage = 3/ std er(3)

= 0.46/0.23 = 2

T-ratio for 30-yr mortgage =  4/ std er(4)

= -0.52/0.40 = -1.3

Thus , when we look at all the t-ratio all the variables are significant except 30-yr mortgage as the t-ratio for this variable is less than 2.00

C. Autocorrelation refers to the presence of serial correlation in the error terms in the multiple linear regression model. Auto correlation refers to a problem where we have a series of data and and we can easily predict the values in the series from the past values and a pattern is to be found in the series, also named as serial correlation.  Autocorrelation is mostly found in time series data. Durbin watson test is used to check whether their is serial correlation or not, as durbin watson produces test statistic ranging from 2 to 4, the closer values are to 2(less autocorrelation) and the closer values are to 4(more autocorrelation).

Null hypothesis - H0: = 0(presence of serial correlation)

Alternate hypothesis - Ha: is not equal to zero

Given in question DW = 2.75 which is closer to 2, thus this regression model has less presence of autocorrelation) and we reject null hypothesis.

D. Multicollinearity refers to the situtation where independent variables are linearly correlated. This means that changes in any one independent variable brings an interlinked shift in the another independent vaiable. This is a big problem as our results would become biased due to presence of multicollinearity and also reduces the statistial power of our regression model. We see potential multicollinearity problem as the standard errors are not too high, but the 15 and 30year mortgage are highly similar variable and shows very high chances of being linearly correlated.