Fulton Fish Example

The following script estimates the reduced form equations using least squares and the demand equation using two-stage least squares for Graddy’s Fulton Fish example.

In the example, ln(quan) and ln(price) are endogenously determined. There are several potential instruments that are available. The variable stormy may be useful in identifying the demand equation. In order for the demand equation to be identified, there must be at least one variable available that effectively influences the supply of fish without affecting its demand. Presumably, stormy weather affects the fishermen’s catch without affecting people’s appetite for fish! Logically, stormy may be a good instrument.

The model of demand includes a set of indicator variables for day of the week. Friday is omitted to avoid the dummy variable trap. These day of week variables are not expected to affect supply; fishermen catch the same amount on average on any working day. Day of the week may affect demand though, since people in some cultures buy more fish on some days than others.

The demand equation is:

ln(quan) = a1 + a2 ln(price) + a3mon + a4tue + a5wed + a6thu + ed (11-5)

Supply is affected by the weather in the previous three days, which is captured in the indicator variable stormy.

ln(quan) = ві + в2 ln(price) + в3 stormy + es (11.6)

In both demand and supply equations, ln(price) is the right-hand side endogenous variable. Iden­tification of the demand equation requires stormy to be significantly correlated with Iprice. This can be determined by looking at the t-ratio in the Iprice reduced form equation.

For supply to be identified, at least one of the day of the week dummy variables (mon tue wed thu) that are excluded from the supply equation, has to be significantly correlated with Iprice in the reduced form. If not, the supply equation cannot be estimated; it is not identified.

Proceeding with the analysis, open the data and estimate the reduced form equations for Iquan and Iprice. Go ahead and conduct the joint test of the day of the week variables using the —quiet option.

1 open "@gretldirdatapoefultonfish. gdt"

2 #Estimate the reduced form equations

3 list days = mon tue wed thu

4 list z = const stormy days

5 ols lquan z

6 omit days —quiet

7 ols lprice z

8 omit days —quiet

Notice how the list command is used. A separate list is created to contain the indicator variables. This allows us to add them as a set to the list of instruments in line 4 and to test their joint significance in the reduced form equation for price in lines 6 and 8. The reduced form results for lquan appear below:

Model 1: OLS estimates using the 111 observations 1-111
Dependent variable: lquan

Variable

Coefficient

Std. Error

t-statistic

p-value

const

8.810

0.147

59.922

0.000

stormy

-0.388

0.144

-2.698

0.008

mon

0.101

0.207

0.489

0.626

tue

-0.485

0.201

-2.410

0.018

wed

-0.553

0.206

-2.688

0.008

thu

0.054

0.201

0.267

0.790

Standard error of residuals (<r) Unadjusted R2 F (5,105) p-value for F()

and the results for Iprice

Model 2: OLS estimates using the 111 observations 1-111
Dependent variable: lprice

Variable

Coefficient

Std. Error

t-statistic

p-value

const

-0.272

0.076

-3.557

0.001

stormy

0.346

0.075

4.639

0.000

mon

-0.113

0.107

-1.052

0.295

tue

-0.041

0.105

-0.394

0.695

wed

-0.012

0.107

-0.111

0.912

thu

0.050

0.104

0.475

0.636

Unadjusted R2

0.178889

F(5, 105)

4.57511

p-value for F()

0.000815589

In the reduced form equation for price, stormy is highly significant with a t-ratio of 4.639. This implies that the demand equation is identified and can be estimated with the data. A joint test of the significance of the daily indicator variables reveals that they are not jointly significant; the F-statistic has a p-value of only 0.65. Since the daily indicators are being used as instruments to estimate supply, the supply structural equation is not identified by the data and can’t be estimated without better variables.

The two-stage least squares estimates of the demand equation are obtained using:

#TSLS estimates of demand tsls lquan const lprice days ; z

to produce the result:

Model 3: TSLS estimates using the 111 observations 1-111
Dependent variable: lquan
Instruments: stormy

Mean of dependent variable S. D. of dependent variable Sum of squared residuals Standard error of residuals (<r) F (5,105) p-value for F()

Hausman test –

Null hypothesis: OLS estimates are consistent Asymptotic test statistic: x2 = 2.4261 with p-value = 0.119329

First-stage F(1,105) = 21.5174

The coefficient on Iprice is negative and significant. It also appears that demand is significantly lower on Tuesday and Wednesday compared to Fridays. The Hausman test for the exogeneity of Iprice is not rejected at 5%. This suggests that least squares might be a suitable means of estimating the parameters in this case. Also, the instruments appear to be sufficiently strong, i. e., the F = 21.51 > 10.

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