Using Critical Value Method To Test Variable Significance And Elasticity Calculation

1. Using the critical value method, test for the individual significance of the coefficient for each of the variables Inc, and Gp at the 5% significance level.

2. Calculate the gas price elasticity of per household gas consumption and income elasticity of per household gas consumption at the sample mean values and interpret them

Model 4: OLS, using observations 1953-2004 (T = 52)

Model: OLS, using observations 1953-2004 (T = 52)

Dependent variable: PerCap

Therefore, PerCapt = 1181.58 + 0.218651Inc_t – 7.87342Gp_t + 1.26823Pnc_t – 18.0738Ps + 75.298T +?_t

Per household gas consumption is positively related to per capital disposable income, inversely related to the price index for gasoline, positively related to the price index of new cars, negatively associated to the aggregate price index of the consumer services, and positively related to time trend. When the price index for gasoline increases, per household gas consumption would decrease and vice versa.

In the same way, when price for new cars rises, the household consumption will also increase, and vice versa. The rise in the aggregate price for consumer services would also decrease household gas consumption, and vice versa. The estimated signs of these coefficients do agree with my expectations. For instance, since cars are a luxury commodity, the price index for new cars would be expected to be inversely related to household gas consumption.  

From the scatter graph above, it is clear that as per capita disposable income (Inc) increases, per household gas consumption among consumer increases, because now consumers have huge disposable incomes to spend on gas. However, any decrease in the per capital disposal increase would decrease household gas consumption. 

Using the critical value method to test individual significance of the coefficients of Inc and Gp at 5% significance level

• Price Index For Gasoline (Gp) Test 

The coefficient Inc is 0.218651

Therefore,

lb = Coefficient (Inc) – Critical value of t = 0.218651 – 2.0129 = -1.79425

ub = Coefficient (Inc) + Critical value of t = 0.218651 + 2.0129 = 2.23155

Thus,

-1.79425≤ t ≤2.23155

The t-test for the regression coefficient is statistically significant, since it lies within the range. 

Per Capital Disposable Income (Inc) Test 

The coefficient Inc is −7.87342

Therefore,

lb = Coefficient (Inc) – Critical value of t = −7.87342 – 2.0129 = -9.88632

ub = Coefficient (Inc) + Critical value of t = −7.87342 + 2.0129 = -5.86052

-9.88632≤ t ≤-5.86052

The t-test for the regression coefficient is statistically significant, since it lies within the range.  

Gas price elasticity of per household gas consumption (PerCap) and income elasticity of per household gas consumption (PerCap) at the sample mean values

Price elasticity of per capita gas consumption

                    Mean         Median        Minimum        Maximum

Inc               16805.         16692.         8685.0         27113.

Gp                 51.343         47.502         16.668         123.90

Pnc                87.567         78.800         44.800         141.70

Ps                 89.777         64.150         19.400         222.80

T                  26.500         26.500         1.0000         52.000

The price elasticity is computed using the following formula
PE = (ΔQuantity/ΔPprice) * (Price/Qquantity)

The (ΔQuantity/ΔPrice) is presented in the above equation by the coefficient – 7.87342.
To determine (Price/Quantity), the mean of price (51.343) is used as well as the mean of Ps which is 89.777.

Therefore we have Price elasticity = – 7.87342 * 51.343/89.777= -4.50277

This means that an increase in the price by 1 unit will decrease quantity demanded by 4.50277 units.

Income elasticity of per capita gas consumption

Income elasticity = (ΔQuantity/ΔPprice) * (Price/Qquantity)

The (ΔQuantity/ΔPrice) is presented in the above equation by the coefficient – 7.87342.
To determine (Price/Quantity), the mean of price (51.343) is used as well as the mean of Ps which is 87.567.

Dependent variable: PerCap

Therefore we have PE = – 7.87342 * 51.343/87.567= -4.616

This means that a decrease in the income by 1 unit will decrease quantity demanded by 4.616 units. 

The null hypothesis here is represented as β4 - 0 against its alternative, which is positive; thus meaning that β4 >0.  Therefore, the test statistic is given as follows:

t= (b4−0)/se (b4) ∼ t46

Provided that β4 - 0 (and that the null hypothesis is true), we use the confidence alpha= 0.05; thus, making the critical value for the one sided alternative (β4>0) equal to 1.686.

 From here, the decision rule is not accepting H0 in favor of the H1 if the computed value of t-statistic is within the rejection region of the test; especially,  if it is greater than 1.686. 

Model 4: OLS, using observations 1953-2004 (T = 52)

Dependent variable: PerCap 

The calculation is given as:

t- (b4−0)/se (b4) - (-18.0738−0)/1.58039 – (-11.4363)

The critical value of t is 2.0129, substituting

= 2.0129 - (-18.0738−0)/ 1.58039- (-18.0738−0)/1.58039 – (-11.4363)

= 2.0129 -6.637509 /13.01669

= 2.0129 -0.509923 = 1.502977  

Therefore, given that this value is within the rejection area, it then means that there is sufficient evidence at the 5 percent level of significance to induce that the null hypothesis is not correct; therefore, the null hypothesis is not accepted at the 5% level of significance.  

(Log-Log Model)

Ln (PerCapt) = α₁ + α₂ ln (Inc_t) + α₃ ln (Gp_t) + α₄ Pnc_t + α₅ Ps_t + ϒT + w_t 

Model 5: OLS, using observations 1953-2004 (T = 52)

Dependent variable: l_PerCap 

Ln (PerCapt) = 3.1243 + 0.563582 ln (Inc_t) -0.0851467 ln (Gp_t) – 0.000164518Pnc_t – 0.00364812 Ps_t + 0.0212205T + w_t

α₁ = A value of 3.1243 represents the unconditional expected mean of log PerCap.

α₂ = A 100 percent change in Inc leads to a 100*0.563582 percentage point change in the probability that PerCab occurs.

α₃ = A 100 percent change in Inc leads to a 100*0.0851467 percentage point change in the probability that PerCab occurs

α₄ = the movement of 0.000164518 from 0 to 1 produces a 100*0.000164518 percent point change in the probability that PerCab occurs

α₅ = the movement of 0.00364812 from 0 to 1 produces a 100*0.00364812 percent point change in the probability that PerCab occurs

To calculate the 99% confidence interval for ln (Gp_t), I will use the following parameters;

Number of predictors = 5

Regression coefficient of ln (Gp_t) = -0.0851

Sample size = 52

Standard error of ln (Gp_t) = 0.01671

Therefore,

99% confidence interval for ln (Gp_t): -0.13000 ≤ α ≤ -0.04020

This part concerned about testing the overall significance of α₃.  As always performed, I  will go ahead and  test  to  establish  if  the  slope  of the parameter  is  different  from  zero.   If it is not, then the variable related to it could not belong in the model.

Thus, the null hypothesis is H0: α₃ – 0 and the alternative hypothesis is H1: α₃≠0. Then the statistic is given as t = (α₃−0)/se (α₃) ∼t46. Moreover, if the null hypothesis is indeed true, and this is computed as t= (α₃−0)/se (α₃) = (−0.0851467−0)/ 0.0167173= -5.093.  Furthermore, the two-sided, alpha= 0.05 critical value is 2.013 and therefore, -5.093 is within the 5% rejection region of this test.  

Given the following values:

1. The per capita disposable income (Inc) = $28000
2. Gas price index (Gp) = 135
3. Price index for new cars (Pnc) = 130
4. Aggregate price index for consumer services (Ps)= 230
5. T=53.

Then,

PerCapt = 3.1243 + 0.563582 ($28000) -0.0851467 (135) – 0.000164518 (130) – 0.00364812 (230) + 0.0212205 (53)

PerCapt = $15772.19

The critical value at 95% confidence interval is 0.0630498

Critical evaluation and comparison the two models to establish which model is best and why.

First model:

PerCapt = 1181.58 + 0.218651Inc_t – 7.87342Gp_t + 1.26823Pnc_t – 18.0738Ps + 75.298T +?_t

Second model:

Ln (PerCapt) = 3.1243 + 0.563582 ln (Inc_t) -0.0851467 ln (Gp_t) – 0.000164518Pnc_t – 0.00364812 Ps_t + 0.0212205T + w_t

A linear equation such as the first model, which specifies a linear relationship among the its variables provides a good description of the economic behaviors compared to a nonlinear model. An alternative approach is considering a linear relationship among log-transformed variables, which is referred to as log-log model – In this model, as seen from above, the dependent variable and some of the explanatory variables are transformed to logarithms.

Sometimes a variable can be logged because it has a long tail (or it is skewed to the right) whereas its logged value may look like it is normally distributed.

Logged variables are not a good move, since there is no econometric reason to make a variable appear normal.

In essence, the only requirement for OLS regression is that the residuals always appear to be normal. In class, we examined plots of residuals as well as log residuals to try to figure which ones looked more normal. And the OLS regression model had looked more normal compared to the log model. The same case is for the above models.

Besides, taking logs also affect the economic interpretations of the variables. Therefore, liner regression is the best, since it does not alter the economic interpretations of the economic variables as compared to log-log model.