Question

Y	PC	PB	YD	TEMP	PRP
39.7	42.3	143.8	50.1	-16	107.8
38.69	49.4	152.2	54.98	-4	134.6
42.02	45.5	145.7	59.72	-24	134
42.71	45.3	145.9	65.17	16	125.4
44.75	49.3	178.8	72.24	5	143.6
48.35	50	222.4	79.67	13	152.5
48.47	53.5	233.6	88.22	21	147.5
50.37	53.8	234.7	97.65	49	161.2
51.52	51.5	238.4	104.26	4	185.6
52.55	56	234.1	111.31	35	179.7
54.61	61.5	235.5	123.19	11	171.4
56.42	56.2	228.6	130.37	4	170.8
57.7	63.1	226.8	136.49	18	188.8
61.94	53.1	238.4	142.41	35	199.4
63.8	62.1	250.3	152.97	46	194
66.88	64.2	265.7	162.57	32	193.5
70.34	60.5	281	171.31	64	224.9
73.26	57.7	288.3	176.09	52	224.2
76.39	59	284.6	184.94	18	209.5
78.27	27.1	293.4	188.72	27	209.1
79.65	26.2	282.9	195.55	48	209.5
79.27	26.9	284.3	202.87	71	206.1
80.61	28	280.2	210.91	36	233.7
83.1	33.2	279.5	219.4	60	245
83.76	33.4	277.1	231.61	89	242.7
88.98	39.5	287.8	239.68	60	241.4
90.08	43	306.4	254.69	62	258.2
89.71	43.4	337.7	262.24	74	269.4
94.37	43.9	331.5	271.45	85	265.8

Download the data set CHICK.DTA   

1. Estimate demand for per capita chicken consumption Y (in pounds) as a linear function of its price PC (in cents), price of its substitute beef PB, and the consumer disposable income YD. Assess the results.
2. Re-estimate the equation without the variable PB. Comment on the results by comparison with those in a. above.
3. Now re-estimate the equation in a. by adding a new variable for temperature TEMP. Comment on the results by comparison with those in a. above.

Answer 1

The required estimations can be done by simple R-commands. The command and output is shown as below. To input the data in R, use the following code.

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> library(readr)
> dat <- read_delim("dat", "\t", escape_double = FALSE, trim_ws = TRUE)

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(a) The regression output would be as below.

> summary(Im(Y - PC + PB + YD, data = dat)) Call: 'lm(formula = Y - PC + PB + YD, data = dat), Residuals: Min 10 Median 30 Max -3.4623 -1.0557 0.1723 0.9764 4.1514 Coefficients: Estimate Std. Error t value Pr(t) (Intercept) 27.68981 2.50058 11.073 3.94e-11 *** PC -0.10976 0.03247 -3.380 0.00238 ** PB 0.03197 0.01720 1.859 0.07486 YD 0.22669 0.01445 15.686 1.90e-14 *** Signif. codes: O'***' 0.001 ***' 0.01 "*" 0.05 ','0.1''i Residual standard error: 1.71 on 25 degrees of freedom Multiple R-squared: 0.9914, Adjusted R-squared: 0.9904 F-statistic: 964.9 on 3 and 25 DF, p-value: < 2.2e-16

Except the variable PB, all the coefficients are individually statistically significant at 1% (highly significant). However, if the significance level is shifted to 10%, then all the coefficients are significant. What can be stated is that the variable PB is less significant than other variables.

The R-squared is high, and F-statistic (for the test of overall significance) is also highly significant (p-value much less than 1%). This means that the model explains the dependent variable very well.

(b) The regression output would be as below.

> summary(lm(Y - PC + YD, data = dat)) Call: ilm(formula = Y - PC + YD, data = dat), Residuals: Min 1Q Median -3.2288 -0.9927 -0.0116 30 Max 0.8335 4.2617 Coefficients: Estimate Std. Error t value Pr( t) (Intercept) 30.678942 .00333 15.31 1.59e-14 *** PC -0.08658 0.03137 -2.76 0.0104 * YD 0.25174 0.00546 46.10 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 .' 0.1'' Residual standard error: 1.789 on 26 degrees of freedom Multiple R-squared: 0.9903, Adjusted R-squared: 0.9895 F-statistic: 1321 on 2 and 26 DF, p-value: < 2.2e-16

In this case, all the coefficients are individually statistically significant at 5%. But at 1%, PC coefficient is not significant (since p-value greater than 1%).

As can be suspected, the R-square and adjusted R-square both decreased as we have removed a variable. But the F-statistic is certainly significant again.

(c) The regression output would be as below.

> summary(lm(Y - PC + PB + YD + TEMP, data = dat)) Call: lm( formula = Y - PC + PB + YD + TEMP, data = dat), Residuals: Min 1Q Median 30 Max -3.5017 -1.0514 0.2721 0.9331 3.5769 PC Coefficients: Estimate Std. Error t value Pr(t) (Intercept) 26.87937 2.65480 10.125 3.86e-10 *** -0.110190 .03256 -3.384 0.00245 ** 1.991 0.05802 YD 0.23169 0.01546 14.990 1.10-13 *** TEMP -0.01935 0.02082 -0.929 0.36204 Signif. codes: ****' 0.001 '**' 0.01 '*' 0.05 ','0.1''1 Residual standard error: 1.715 on 24 degrees of freedom Multiple R-squared: 0.9917, Adjusted R-squared: 0.9904, F-statistic: 719.9 on 4 and 24 DF, p-value: <2.2e-16

The coefficients of variable PB and TEMP are not statistically significant at 1% or even at 5%. Also, TEMP is not at all significant at 10% too. Rest of the variables are highly significant as before.

The R-square marginally increased, but the adjusted R-square is same as before. The F-statistic is also significant, as before.

All that can be said is that, the TEMP and PB variable must be tested via restricted and unrestricted regression, to be included in the model.