Question

show all work include  any Stata work

Use the data set chick6.xls in Moodle to answer the following:

In the chick6 data set y=per capita chicken consumption, pounds per year; yd= disposable income per capita, hundreds of dollars; pc=price of chicken, cents per pound; pb=price of beef, cents per pound.

Before beginning you will want to inform STATA that this dataset are in the time-series format. To do this, generate a variable that identifies time.  In STATA, type generate time=_n.   Then type tsset time.

1. Estimate the linear function for the demand for chicken using price of chicken (pc), price of beef (pb), and disposable income (yd) as independent variables. What is the price elasticity of chicken? Is it inelastic or elastic? Explain
2. In STATA, calculate the correlation coefficients and VIF for the independent variables. Are there any signs of multicollinearity?  Why or why not?  Be sure to indicate in which variables multicollinearity is likely an issue.Explain.
3. Formulate the hypothesis for the Ramsey RESET test. Then, perform the Ramsey RESET test in STATA. Are there any signs of biased results? Why or why not?  Explain.
4. Estimate the lin-log model of the equation in (a). Which model, the linear or the lin-log fits better? Explain.
5. Estimate the log-log model of the equation in (a). What is the price elasticity of demand? Does this model fit better than the linear or lin-log? Explain.  

Chick6 data

OBS	YEAR	Y	PC	PB	YD	TEMP	PRP
1	1974	39.7	42.3	143.8	50.1	-16	107.8
2	1975	38.69	49.4	152.2	54.98	-4	134.6
3	1976	42.02	45.5	145.7	59.72	-24	134
4	1977	42.71	45.3	145.9	65.17	16	125.4
5	1978	44.75	49.3	178.8	72.24	5	143.6
6	1979	48.35	50	222.4	79.67	13	152.5
7	1980	48.47	53.5	233.6	88.22	21	147.5
8	1981	50.37	53.8	234.7	97.65	49	161.2
9	1982	51.52	51.5	238.4	104.26	4	185.6
10	1983	52.55	56	234.1	111.31	35	179.7
11	1984	54.61	61.5	235.5	123.19	11	171.4
12	1985	56.42	56.2	228.6	130.37	4	170.8
13	1986	57.7	63.1	226.8	136.49	18	188.8
14	1987	61.94	53.1	238.4	142.41	35	199.4
15	1988	63.8	62.1	250.3	152.97	46	194
16	1989	66.88	64.2	265.7	162.57	32	193.5
17	1990	70.34	60.5	281	171.31	64	224.9
18	1991	73.26	57.7	288.3	176.09	52	224.2
19	1992	76.39	59	284.6	184.94	18	209.5
20	1993	78.27	27.1	293.4	188.72	27	209.1
21	1994	79.65	26.2	282.9	195.55	48	209.5
22	1995	79.27	26.9	284.3	202.87	71	206.1
23	1996	80.61	28	280.2	210.91	36	233.7
24	1997	83.1	33.2	279.5	219.4	60	245
25	1998	83.76	33.4	277.1	231.61	89	242.7
26	1999	88.98	39.5	287.8	239.68	60	241.4
27	2000	90.08	43	306.4	254.69	62	258.2
28	2001	89.71	43.4	337.7	262.24	74	269.4
29	2002	94.37	43.9	331.5	271.45	85	265.8

Answer 1

a) syntax

set obs 29
number of observations (_N) was 29, now 29

. generate time =_n

. tsset time
time variable: time, 1 to 29
delta: 1 unit

. reg Y PC PB YD

Regression result

Source | SS df MS Number of obs = 29
-------------+---------------------------------- F(3, 25) = 964.88
Model | 8465.4915 3 2821.8305 Prob > F = 0.0000
Residual | 73.113677 25 2.92454708 R-squared = 0.9914
-------------+---------------------------------- Adj R-squared = 0.9904
Total | 8538.60518 28 304.950185 Root MSE = 1.7101

------------------------------------------------------------------------------
Y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
PC | -.1097636 .0324726 -3.38 0.002 -.1766421 -.042885
PB | .0319748 .0172014 1.86 0.075 -.0034522 .0674017
YD | .2266944 .0144516 15.69 0.000 .1969308 .256458
_cons | 27.68981 2.500578 11.07 0.000 22.53977 32.83985
-----------------------------------------------------------------------------

For elasticity

Syntax

margins, eyex(PC) atmeans

Result

Conditional marginal effects Number of obs =
> 29
Model VCE : OLS

Expression : Linear prediction, predict()
ey/ex w.r.t. : PC
at : PC = 47.53793 (mean)
PB = 247.9172 (mean)
YD = 153.1303 (mean)

--------------------------------------------------------------------
> ----------
| Delta-method
| ey/ex Std. Err. t P>|t| [95% Conf.
> Interval]
-------------+------------------------------------------------------
> ----------
PC | -.0801369 .023711 -3.38 0.002 -.1289706
> -.0313031
--------------------------------------------------------------------

The price elasticity of chicken is Inelastic if we look at the absolute value of ey/ex the value is less than 1 which Imply inelastic demand.

b)Syntax

corr ( Y PC PB YD)
(obs=29)

| Y PC PB YD
-------------+------------------------------------
Y | 1.0000
PC | -0.4399 1.0000
PB | 0.9231 -0.2291 1.0000
YD | 0.9937 -0.3932 0.9247 1.0000

From the above output result, it is clearly visible that there is a problem of multicollinearity between the variables PB and Y, as well as in between YD and Y also YD and PB

The syntax for the variance inflation factor.
vif

Variable | VIF 1/VIF
-------------+----------------------
YD | 9.07 0.110276
PB | 8.09 0.123596
PC | 1.39 0.720682
-------------+----------------------
Mean VIF | 6.18

Two variables have VIF greater than 4 that is YD and PB. Hence we can conclude that there is a problem of multicollinearity.

c) The null hypothesis is that the model has no omitted variables; the alternative is that there is.

The following is the Syntax for RAMSEY RESET TEST

estat ovtest

Ramsey RESET test using powers of the fitted values of Y
Ho: model has no omitted variables
F(3, 21) = 5.58
Prob > F = 0.0056

At 5% level of significance we can reject the null Hypothesis and we can conclude that the model has omitted variables hence Significant.

d) First, we will generate the log variable for lin-log model and after generating the variable we will regress the newly generated variables with the dependent variable.

The following are the syntax and teh output file.

gen lpc = log(PC)

. gen lpb = log(PB)

. gen lyd = log(YD)

. reg Y lpc lpb lyd

Source | SS df MS Number of obs = 29
-------------+---------------------------------- F(3, 25) = 214.96
Model | 8219.94858 3 2739.98286 Prob > F = 0.0000
Residual | 318.656601 25 12.746264 R-squared = 0.9627
-------------+---------------------------------- Adj R-squared = 0.9582
Total | 8538.60518 28 304.950185 Root MSE = 3.5702

------------------------------------------------------------------------------
Y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lpc | -9.028432 2.720089 -3.32 0.003 -14.63056 -3.426303
lpb | -8.940055 8.807944 -1.01 0.320 -27.08035 9.200245
lyd | 35.9398 4.374833 8.22 0.000 26.92966 44.94994
_cons | -28.1116 28.50739 -0.99 0.334 -86.82367 30.60047

The linear model is better because the R squared value is 99% which is greater than the lin-log model and hence is the best model.

d) For log-log model first, we will generate the dependent variable in form of log with the help of "generate" syntax.

gen lY =log(Y)

now we will regress the double log model with lY as the dependent variable.

reg lY lpc lpb lyd

Source | SS df MS Number of obs = 29
-------------+---------------------------------- F(3, 25) = 531.05
Model | 2.13648586 3 .712161955 Prob > F = 0.0000
Residual | .033526309 25 .001341052 R-squared = 0.9846
-------------+---------------------------------- Adj R-squared = 0.9827
Total | 2.17001217 28 .077500435 Root MSE = .03662

------------------------------------------------------------------------------
lY | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lpc | -.1028254 .0279007 -3.69 0.001 -.1602879 -.0453629
lpb | -.0334645 .0903454 -0.37 0.714 -.2195343 .1526053
lyd | .5409689 .0448738 12.06 0.000 .4485496 .6333883
_cons | 2.055127 .2924077 7.03 0.000 1.452902 2.657352

The coefficient of the log-log model gives the value of elasticity. Therefore from the output file we can simply see the elasticity value.

We cannot compare the log-log model with the lin-log model and the linear equation model because the dependent variables are not the same and hence we can not make use of the R² that defines the fitness of the model.