Question

3300 Econometics HW Set 1

DATE	Cons.	Disp.Icome
2015-01-01	$ 11,788.36	$ 13,226.57
2015-04-01	$ 11,887.54	$ 13,327.81
2015-07-01	$ 11,971.95	$ 13,440.36
2015-10-01	$ 12,039.65	$ 13,471.39
2016-01-01	$ 12,111.78	$ 13,562.27
2016-04-01	$ 12,214.10	$ 13,541.45
2016-07-01	$ 12,294.30	$ 13,592.92
2016-10-01	$ 12,372.73	$ 13,685.36
2017-01-01	$ 12,427.65	$ 13,835.34
2017-04-01	$ 12,515.86	$ 13,909.77
2017-07-01	$ 12,584.91	$ 13,986.19
2017-10-01	$ 12,706.37	$ 14,065.92
2018-01-01	$ 12,722.84	$ 14,219.83
2018-04-01	$ 12,842.02	$ 14,306.61
2018-07-01	$ 12,968.54	$ 14,393.59

The data given in the data file in the Consumption file represent the real private consumption of the USA from Quarter I 2005 to III Quarter 2018.

Similarly, the Real Disposable Income is provided over the same time span.

Set up a regression that relates the dependent variable(Y) to the independent variable(X).

Derive Manually the coefficients of the regression. (Intercept(b1) and slope(b2)).

State the Regression equation.

Interpret the meaning of the slopes b2, in this problem.

Derive the Correlation Coefficient R^2

Derive the Standard Error of the regression

Derive the standard error of the Intercept (b1) and the standard error of the Slope (b2).

Derive the t values of the coefficients

Construct a 95% confidence interval for b1 and b2

Use a two tail α=5% level of significance, to test the confidence intervals for the slope(b2).

(Hint: All the formulas required to answer the questions are cited in chapters 2 and 3 of the textbooks. Use also the notes from the lectures).

Answer 1

The regression equation is defined as,

The least square estimate of intercept and slope are,

	Consumption, X	X^2	Disp.Icome, Y	X*Y	Y^2
	11,788.36	13,89,65,431.49	13,226.57	155919568.7	17,49,42,153.9649
	11,887.54	14,13,13,607.25	13,327.81	158434874.5	17,76,30,519.3961
	11,971.95	14,33,27,586.80	13,440.36	160907317.9	18,06,43,276.9296
	12,039.65	14,49,53,172.12	13,471.39	162190820.6	18,14,78,348.5321
	12,111.78	14,66,95,214.77	13,562.27	164263230.5	18,39,35,167.5529
	12,214.10	14,91,84,238.81	13,541.45	165396624.4	18,33,70,868.1025
	12,294.30	15,11,49,812.49	13,592.92	167115436.4	18,47,67,474.1264
	12,372.73	15,30,84,447.65	13,685.36	169325264.2	18,72,89,078.3296
	12,427.65	15,44,46,484.52	13,835.34	171940763.2	19,14,16,632.9156
	12,515.86	15,66,46,751.54	13,909.77	174092734	19,34,81,701.4529
	12,584.91	15,83,79,959.71	13,986.19	176014942.4	19,56,13,510.7161
	12,706.37	16,14,51,838.58	14,065.92	178726783.9	19,78,50,105.4464
	12,722.84	16,18,70,657.67	14,219.83	180916621.9	20,22,03,565.2289
	12,842.02	16,49,17,477.68	14,306.61	183725771.8	20,46,79,089.6921
	12,968.54	16,81,83,029.73	14,393.59	186663847.7	20,71,75,433.0881
SUM	1,85,448.60	2,29,45,69,710.81	2,06,565.38	2,55,56,34,602.04	2,84,64,76,925.47

Form the data values, the values are calculated as,

Now, for one unit increase in consumption, Disp.Income will increase by approximately 0.9962.

The correlation coefficient is obtained using the formula

Now, the R^2 value is,

The standard error of the regression is obtained using the formula,

Where, is estimated using the regression equation for X values,

Consumption, X	Disp.Icome, Y	Yhat	(Y-Yhat)^2
11,788.36	13,226.57	13198.33	797.57
11,887.54	13,327.81	13297.13	941.14
11,971.95	13,440.36	13381.22	3497.37
12,039.65	13,471.39	13448.66	516.46
12,111.78	13,562.27	13520.52	1743.03
12,214.10	13,541.45	13622.45	6561.28
12,294.30	13,592.92	13702.35	11974.30
12,372.73	13,685.36	13780.48	9047.68
12,427.65	13,835.34	13835.19	0.02
12,515.86	13,909.77	13923.07	176.78
12,584.91	13,986.19	13991.85	32.07
12,706.37	14,065.92	14112.85	2202.63
12,722.84	14,219.83	14129.26	8202.99
12,842.02	14,306.61	14247.99	3436.66
12,968.54	14,393.59	14374.03	382.73
		SUM	49512.73

Similarly, the standard error for intercept and slope are obtained using the formula,

T-value for intercept and slope are obtained using the formula,

Where t value is obtained using the t distribution table for and degree of freedom = n - 2 = 15 - 2 = 13

The P-value for the significance for the 95% confidence interval is obtained using the t distribution table for t = 21.8018 and df = 13,

P-value = 0.0000 < 0.05 at 5% significance level. Hence the slope is significant at % significant level.