Question

Refer to the the annual data for the manufacturing sector for 24 years, Eden obtained following
regression results:
log Y = - 0.02 - 0.12 log K + 0.80 log L + 0.05t
se = (2.38) (0.34) (0.009) (0.020) (1)
?2 = 0.99 F = 200
where Y = index of real output, K = index of real capital input, L = index of real labor input,
t = time or trend.
Using the same data, he also obtained the following regression:
log (Y/L) = 0.10 + 0.14 log (K/L) + 0.06t
se = (0.02) (0.02) (0.006) (2)
?2 = 0.85 F = 19.5

a) Is there multicollinearity in regression (1) (at 5%)? How do you know?
b) In regression (1), what is the a prior sign of log K? Do the results conform to this
expectation? Why or why not?
c) Interpret regression (1). What is the role of the trend variable in this regression?
d) What is the logic behind estimating regression (2)?
e) If there was multicollinearity in regression (1), has that been reduced by regression (2)? How
do you know?

Answer 1

1. Log Y = - 0.02 - 0.12 log K + 0.80 log L + 0.05t
se = (2.38) (0.34) (0.009) (0.020) (1)
?2 = 0.99 F = 200

This is regression 1.

Consequences of multicollinearity is that the coefficients may have the wrong sign Or the R² is high but few significant t ratios i. e few coefficients are not significant despite higher joint significance.

We can see that regression 1 suffers from multicollinearity due to :

1. Wrong sign of coefficient (log K) - according to the theory, as capital increases, the output should also increase and hence there must be a positive relationship between log Y and log K but we see that the coefficient of logK is negative.

2. The R² is very high at 0.99 indicting that 99% of the variation in Y is explained by explanatory variables which is unlikely. Regarding individual significance :

T ratios:

Log K: t= -0. 12/0.34 = -0. 3529

We know that we reject the null hypothesis that the coefficient is not significant if:

t calculated> t critical

Or -t actual<- t critical.

The t critical is 2.08. Since -0. 35> -2. 08, we accept the null hypothesis and hence the coefficient is not significant in log K

2. Log L: t= 0.8/0.009 = 88.88

88.88> 2.08, we reject the null. Hence the Coefficient is significant  

3. Trend variable : t= 0.05/0.02= 2.5

We reject the null and hence the coefficient is significant.

We conclude that the R squared is highly significant whereas the coefficient of log K is insignificant. These reasons indicate the presence of multicollinearity.

B) The prior sign of log K must be positive as capital increases the output will also increase . The results show a negative relationship probably due to presence of multicollinearity.

C) R² is 0.99 indicating that 99% of total variation in output is explained by the explanatory variables.

Interpreting the coefficients:

1. When capital increases by 1%, on an average the output decreses by 0.12% , sweeping the linear influence of other explanatory variables.

2. When labour increase by 1% , on an average the output increases by 0.8% , sweeping out the linear influence of other explanatory variables

3. Trend variable shows the rate of growth of output. It indicates that- On an average, the output grows at the rate of 5% ( 0.05*100) per year.

D) in regression 2, we have taken variables in person capita term, ie. We have transformed the nominal variables into real variables. This will help reduce the problem of multicollinearity and remove possibility of linear relationship between variables  

E) yes, the multicollinearity problem has been solved by second regression. We can see the sign of coefficient of log K/L os positive as expected. The R squared is 0.85 , decresed from a very high 0.99 and all coeficients are now significant.