Question

Consider the Cobb-Douglas production function Y = e^b0 K ^b1 L^b2 e^ui where Y, K and L denote real output, real capital input, and real labor input, respectively. The data for estimating the parameters of the production function are given in the Excel data file productionfunction.xls.

1. Perform a logarithmic transformation of the production function to linearity so that it can be estimated by OLS.
2. Compute the correlation coefficient between income lnK and lnL and comment on the potential for multicollinearity.
3. Obtain OLS estimates of the model ln Yi = b0 + b1 ln Ki + ui   i=1,2,...,24 and their t-ratios and comment on the significance of the slope coefficient. Use 5 percent level of significance. Does the estimated slope coefficient make economic sense?
4. Obtain OLS estimates of the model ln Yi = b0 + b1 ln Li + ui i=1,2,...,24 and their t-ratios and comment on the significance of the slope coefficient. Use 5 percent level of significance. Does the slope coefficient make economic sense?
5. Obtain OLS estimates of the model ln Yi = b0 + b1 ln Ki + b2 ln Li + ui i=1,2,...,24 and their t-ratios and comment on the significance of the slope coefficients. Use 5 percent level of significance. Do the slope coefficients make economic sense?
6. By comparing the results in parts (c), (d) and (e), what is the impact of multicollinearity on the signs and significance of the coefficients?

Production Function:

Y	K	L
100	100	100
101	107	105
112	114	110
122	122	118
124	131	123
122	138	116
143	149	125
152	163	133
151	176	138
126	185	121
155	198	140
159	208	144
153	216	145
177	226	152
184	236	154
169	244	149
189	266	154
225	298	182
227	335	196
223	366	200
218	387	193
231	407	193
179	417	147
240	431	161

Answer 1

(a) The logarithmic transformation of the variables are done as follows. The left hand side table shows the calculated values of ln(Y), ln(K) and ln(L), whereas the right hand side table shows the formula view of the spreadsheet.

(b) The correlation coefficient between ln(K) and ln(L) is calculated to be 0.91 (upto 2 decimal places), which means a variance inflation factor (VIF) = 1/ (1-0.912) = 5.79 (upto 2 decimal places). In a strict sense, VIF above 2.5 raises a concern, however, some researchers prefer to consider VIF < 10. Depending upon the objective, the level of 5.79 may raise some concern of multicollinearity, however, it may not be severe.

(c) The regression results for the equation

is shown below.

The t-value of the slope coefficient is is 13.668 and the corresponding p-value is very low (less than 0.05). Hence, the coefficient is statistically significant at 5% level of significance.

(d)

The regression results for the equation

is shown below.

The t-value of the slope coefficient is is 17.115 and the corresponding p-value is very low (less than 0.05). Hence, the coefficient is statistically significant at 5% level of significance.

(e)

The regression results for the equation

is shown below.

The t-values of the slope coefficients are respectively 3.668 and 5.565 and the corresponding p-values are also low enough to make sure that both the coefficients are statistically significant at 5% level of significance.

As observed, due to collinearity, the respective standard errors are more than their individual regressions. This reduces th t-values. A higher level of correlation may have made the coefficients insignificant. However, in this case, both are significant at 5% level of significance.

Since the coefficients indicate respective outout elasticities, they looks economically sensible as they are less than unity individually. Also it may be observed that the sum of the coefficients exceed 1, which measn that the production function exhibits increasing returns to scale.

(f) The signs are intact, i.e., both the coefficients remain positive. The significance of both the coefficients reduce to some extent. However, they are still significant at 5%.