# A demand curve for wine in ruritania

Estimating an appropriate demand function for Wine 6 and interpreting the results

## Introduction

In this report, I will estimate and analyse a demand curve for wine in Ruritania between 1980 and 2009, with the data produced quarterly. I will use the framework of the General Linear Model to run various regressions, estimations and tests. I will identify the correct functional form and then I will identify significant, explanatory variables to ascertain the preferred model. Demand homogeneity will be tested and I will use Slutsky’s equation to determine the income and substitution effects. From there I will interpret the model and run additional tests to identify seasonal and structural changes in the wine market.

The volume of wine consumed has shown worldwide growth between 2002 and 2010 (The Wine Institute, 2009). Wine is a luxury good and thus has a high elasticity of demand (Ohidul Haque, 2005). The elasticity of demand for wine can be attributed to a vast number of factors; however this report will investigate the 10 independent variables found in the table below. The demand function for wine will be constructed using the significant variables from the data set. Influences such as social demographics and cultural differences will be ignored and so I have assumed the population of Ruritania to be essentially identical.

Table 1

## Variable

## Description

QWINE (6)

Quantity demanded of wine

PMTFH

Price of meat and fish

PFTVG

Price of fruit and vegetables

PTEA

Price of tea

PCOFF

Price of coffee

PBEER

Price of beer

PWINE

Price of wine

PLEIS

Price of leisure

PTRAV

Price of travel

PALLOTH

Index of all other prices

INCOME

Income

## Identification of functional form

To obtain a demand curve, it is important to identify the correct functional form as ‘nothing says the relationship must be a straight line’. I will look at four common functional forms are test their statistical efficacy. (Carter Hill, Griffiths, & Guay, 2008)

Linear model:

QWINE= 0 + 1PMTFH + 2PFTVG + 3PTEA + 4PCOFF + 5PBEER + 6PWINE + 7PLEIS + 8PTRAV + 9PALLOTH + 10 INCOME

Log-Log model:

LQWINE= 0 + 1LPMTFH + 2LPFTVG + 3LPTEA + 4LPCOFF + 5LPBEER + 6LPWINE + 7LPLEIS + 8LPTRAV + 9L PALLOTH + 10LINCOME

Log-Linear Model:

LQWINE= 0 + 1PMTFH + 2PFTVG + 3PTEA + 4PCOFF + 5PBEER + 6PWINE + 7PLEIS + 8PTRAV + 9PALLOTH + 10INCOME

Linear-Log Model:

QWINE= 0 + 1LPMTFH + 2LPFTVG + 3LPTEA + 4LPCOFF + 5LPBEER + 6LPWINE + 7LPLEIS + 8LPTRAV + 9LPALLOTH + 10LINCOME

‘Economists use the ... log-log model specification ... frequently’ as the ‘parameter[s] [1 to 10] are the elasticity of y with respect to x’ (Carter Hill, Griffiths, & Guay, 2008). Initially, this suggests that the log-log model may be the preferred model as this report focuses on elasticity.

I will use PCGive to run several tests outlined in table 2 on each of the functional forms. The results of the tests can be seen, along with the critical values in table 3.

Table 2 (Gujurati, 2003) (Baiocchi, 2010)

## Diagnostic Test

## Test factor

## Hypotheses

Ramsey’s RESET (RR)

General Specification error

H0: Test statistic < Critical Value

Model correctly specified

H1: Test statistic > Critical Value

Model incorrectly specified

Jarque-Bera (JB)

Normality (Asymptotic)

H0: Test statistic < Critical Value

Disturbance term normally distributed

H1: Test statistic > Critical Value

Disturbance term not normally distributed

F-test

Overall significance of multiple regression

H0: Test statistic < Critical Value

Model overall insignificant

H1: Test statistic > Critical Value

Model overall significant

White’s Test (HS)

Heteroskedasticity

H0: Test statistic < Critical Value

Homoskedasticity

H1: Test statistic > Critical Value

Heteroskedasticity

Breusch-Godfrey (BG)

Auto-correlation (incl. lag dependent variables)

H0: Test statistic < Critical Value

No autocorrelation in data

H1: Test statistic > Critical Value

Autocorrelation in data

Durbin-Watson (DW)

Auto-correlation (excl. lag dependent variables)

H0: Test statistic < dL

Positive autocorrelation in data

H1: dU > Test statistic > dL

Indecision

H2: 2 > Test statistic > dU

No autocorrelation in data

H3: Test statistic > 2

Redo test for negative autocorrelation using (4-d) as test statistic

R²

Percentage of data explained by the regression

n/a

Table 3

## Diagnostic Test

## Critical Value

## (5% significance)

## Linear Form

## Log-Log Form

## Log-Linear Form

## Linear-Log Form

Ramsey's RESET

3.92

16.935

2.1076

17.097

182.24

Jarque-Bera

5.99147

52.376

0.44649

1.7793

9.9281

F-test

1.91

61.07

225.7

167.9

65.88

White’s Test

1.64

7.4546

0.84785

1.1955

2.1019

Breusch-Godfrey

2.29

27.358

1.0791

2.1736

18.170

Durbin-Watson

dL = 1.462

dU = 1.898

0.929

2.17

1.81

1.01

R²

n/a

0.848538

0.953936

0.93905

0.85804

xxxxxx - Pass, xxxxxx - Fail, xxxxxx - Indecision

Both the linear and the linear-log functional forms fail Ramsey’s Reset (RR) test, the Jarque-Bera (JB) test, White’s test, the Durbin-Watson (DW) test and both models only explain around 85% of the data which is comparatively low. Despite passing the F-Test, it can be safely assumed from the results above that the demand function for wine is neither in a linear or linear-log form.

The remaining two models, the log-log and log linear forms, both pass the JB, White’s, BG and F test however the log-log form passes these tests more comfortably in each instance. The log-linear form fails the RR test by a significant margin. This is a significant result considering that the RR test is used to check for correct model specification.

The log-log model produced a value of 2.17 for the DW test, so when checking for negative autocorrelation, it falls in the region of indecision at the 5% significance level. The log-linear model also falls in the region of indecision. At the 1% significance level, dL = 1.335 and dU = 1.765 and both models pass the test and therefore show little sign of autocorrelation. This is strengthened by a pass for both models in the BG test. The log-log model explains 95.4% of the data, whilst the log-linear model explains 93.9% of the data.

The log-log constant elasticity functional form passes every test in a more numerically convincing way than the log-linear functional form and crucially, the log-linear form fails the RR test. For these reasons, I will choose the log-log functional form as the most appropriate in modelling the demand function for wine, thus confirming my initial prediction.

## Finding the preferred model

Pan, Fang and Malaga used a censored demand system approach in looking at beer and wine consumption in China between 1993 and 1998. They found that ‘wine is still a luxury good in China’ and that ‘there is substitution ... between wine and beer’ (Pan, Fang, & Malaga, 2006). This is consistent with my assumption that wine is a luxury good and has provided a predictable insight into the relationship between wine and beer. Similar conclusions were also drawn in a paper by Salisu and Balasubramanyam which studies the elasticities of wine, spirits and beer in response to the announcement of the 2004 budget in the UK. Interestingly, beer and spirits were found to be consistent with demand homogeneity, however ‘the wine data may not be consistent with the homogeneity restriction’ (Salisu & Balasubramanyam, 1997).

Having identified the functional form, I will now identify any variables that need to be removed in order to obtain the most accurate model. I will use the t-test to eliminate insignificant variables and then use the F-test to test the overall significance of the revised model with insignificant variables removed. An individual variable is statistically insignificant at the 5% significance level when the magnitude of the test value is less than 1.98.

I will retain the constant 0 regardless of its individual significance. This is to ensure that the model remains unbiased, as according to Brooks (Brooks, 2008); removal of the constant could lead to a biased model.

Table 4

## Parameter

## Coefficient

## Estimated Standard Error

## T-value

Critical Value = ±1.98

(5% significance)

Constant

β0 = 5.33568

3.387

1.58

LPMTFH

β1 = -1.23419

0.228

-5.41

LPFTVG

β2 = -0.889495

0.3685

-2.41

LPTEA

β3 = -0.549482

0.3298

-1.67

LPCOFF

β4 = 0.135631

0.3357

0.404

LPBEER

β5 = 2.40214

0.2981

8.06

LPWINE

β6 = -0.966195

0.4002

-2.41

LPLEIS

β7 = -0.32996

0.221

-1.49

LPTRAV

β8 = -0.624662

0.1955

-3.19

LPALLOTH

β9 = -0.121975

0.68

-0.179

LINCOME

β10 = 1.42556

0.2742

5.2

xxxxxx - Significant, xxxxxx – Not Significant, xxxxxx – n/a

From the results above it can be seen that LPCOFF and LPALLOTH are clearly insignificant with test values of 0.404 and -0.179. I will retain LPTEA and LLEIS as they have scores of -1.67 and -1.49 respectively, which are close to the critical value. I will now run a regression of the new model with the variables LPCOFF and LALLOTH removed. The results are displayed in the table below.

Table 5

## Parameter

## Coefficient

## Estimated Standard Error

## T-value

Critical Value = ±1.98

(5% significance)

Constant

β0 = 5.14591

2.398

2.15

LPMTFH

β1 = -1.24566

0.2231

-5.58

LPFTVG

β2 = -0.951713

0.2675

-3.56

LPTEA

β3 = -0.633295

0.2296

-2.76

LPBEER

β5 = 2.39716

0.1929

12.4

LPWINE

β6 = -0.918221

0.3572

-2.57

LPLEIS

β7 = -0. 345367

0.1995

-1.73

LPTRAV

β8 = -0. 650645

0.1751

-3.72

LINCOME

β10 = 1. 41051

0.2597

5.43

xxxxxx - Significant, xxxxxx – Not Significant, xxxxxx – n/a

First, I will run an F-test to ensure that the removal of LPCOFF and LALLOTH has improved the overall significance of the model. The value for R2 in this model is 0.953866. The test statistic is

Where r = number of restrictions, n = number of observations, k = number of parameters in unrestricted model. The critical value for this F-test at 5% significance is , with

H0: β4 = β9 = 0

H1: Not all equal to 0.

The test statistic is computed as F = 0.08358 (4 d.p.) < 3.07. Therefore the null hypothesis is accepted and the variables LPCOFF and LALLOTH are removed. I will now run a regression with LPLEIS removed as its t-value of -1.73 is greater than -1.98 and so it is still individually insignificant. The results are displayed in table 6 below.

Table 6

## Parameter

## Coefficient

## Estimated Standard Error

## T-value

Critical Value = ±1.98

(5% significance)

Constant

β0 = 2.31584

1.769

1.31

LPMTFH

β1 = -1.25116

0.2251

-5.56

LPFTVG

β2 = -0.893851

0.2677

-3.34

LPTEA

β3 = -0.459005

0.2082

-2.21

LPBEER

β5 = 2.51069

0.183

13.7

LPWINE

β6 = -0.796731

0.3533

-2.25

LPTRAV

β8 = -0.706289

0.1736

-4.07

LINCOME

β10 = 1.52185

0.2539

5.99

xxxxxx - Significant, xxxxxx – Not Significant, xxxxxx – n/a

To test how the removal of LPLEIS has affected the model, I will run another F-test. The R2 for this model is 0.952621. The critical value for this F-test at 5% significance is , with

H0: β7 = 0

H1: β7 ≠ 0

The test statistic is computed as F = 3.0225 (4 d.p.) < 3.92. Therefore the null hypothesis is accepted and the variable LPLEIS is removed. The new restricted model can be found below:

LQWINE = 2.31584 – 1.25116LPMTFH – 0.893851LPFTVG – 0.459005LPTEA + 2.51069LPBEER –

0.796731LPWINE – 0.706289LPTRAV + 1.52185LINCOME

## Additional Tests

I will now run tests for structural breaks, seasonality and lags.

Structural Breaks

The graph below shows LQWINE plotted against time. If a structural break exists, then it is most likely to have occurred in the first quarter of 1998. The quantity of wine consumed after 1998 seems to rise. This may be consistent with a tax freeze on spirits in Ruritania, as was seen in the UK during that time (Harpers, 2008).

I will run a chow test to check for a structural break in the first quarter of 1998.

Table 7

## Time period

## Number of observations

## RSS

1980 - 1997

72

14.4442324

1998 - 2009

48

9.95373194

1980 - 2009

120

26.1189223

The test statistic for the chow test can be found below:

n1 and n2 are the number of observations for the first and second periods respectively and k is the number of parameters estimated. The critical value at the 5% level of significance is with,

H0: No structural change

H1: There is a structural change.

The test statistic is computed as 1.0681 (4 d.p.) < 2.09. Therefore, as the test statistic is less than the critical value, I will reject H1 and accept the null hypothesis that there is no evidence of a structural change at the 5% level of significance.

Seasonal Dummies

As wine is produced on a seasonal basis, one might expect there to be changes in consumption at different times of year. In order to avoid the dummy variable trap, I will use just three dummy variables, use the fourth quarter as the benchmark and then keep the intercept. ‘Another way to avoid the dummy variable trap is to omit the intercept from the model. This choice is less desirable than omitting one dummy variable, since omitting the intercept alters several of the key numerical properties of the least squares estimator’ (Carter Hill, Griffiths, & Guay, 2008). I will run a regression of the model rewritten below and the dummies will be t-tested:

LQWINE= 0 + 1LPMTFH + 2LPFTVG + 3LPTEA + 5LPBEER + 6LPWINE + 8LPTRAV + 10LINCOME + 11D1 + 12D2 + 13D3

Table 8

## Parameter

## Quarter

## Coefficient

## Estimated Standard Error

## T-value

Critical Value = ±1.980

(5% Significance)

Dummy 1 (D1)

1st Quarter

11 = 0.0173776

0.1284

0.135

Dummy 2 (D2)

2nd Quarter

12 = -0.0573845

0.1286

-0.446

Dummy 3 (D3)

3rd Quarter

13 = -0.0476386

0.1282

-0.372

xxxxxx - Significant, xxxxxx – Not Significant

The other variables remained significant and largely unchanged with the inclusion of seasonal dummies. The R2 statistic is 0.952828 but the dummies are all individually insignificant. However before removing them I will run an F-test to test their combined significance. The critical value for this F-test at 5% significance is , with

H0: β11 = β12 = β13 = 0

H1: Not all equal to zero.

The test statistic is computed as F = 0.1609 (4 d.p.) < 2.68. Therefore the null hypothesis is accepted and all the dummies are removed. From the diagnostic tests it can be seen that the demand for wine shows no signs of seasonality at the 5% level of significance.

Test for lags

The demand for wine may be dependent on previous demand for wine. This could be due to habitual or even addicted consumers buying a certain amount of a good as that is what they bought last time. To test the affect this may have on the model, I will include the variables LQWINE_1, LQWINE_2, LQWINE_3, LQWINE_4 which will test for a lag in the previous year. I will run a regression of the model rewritten below and the lags will be t-tested:

LQWINE= 0 + 1LPMTFH + 2LPFTVG + 3LPTEA + 5LPBEER + 6LPWINE + 8LPTRAV + 10LINCOME + 11 LQWINE_1 + 12LQWINE_2 + 13LQWINE_3 + 14LQWINE_4

Table 9

## Parameter

## Quarter

## Coefficient

## Estimated Standard Error

## T-value

Critical Value = ±1.980

(5% Significance)

LQWINE_1

1st Quarter Lag

11 = -0.119182

0.08744

-1.36

LQWINE_2

2nd Quarter Lag

12 = 0.149835

0.08278

1.81

LQWINE_3

3rd Quarter Lag

13 = 0.0410803

0.07757

0.53

LQWINE_4

4th Quarter Lag

14 = -0.0324007

0.07614

-0.426

xxxxxx - Significant, xxxxxx – Not Significant

The other variables remained significant and largely unchanged with the inclusion of quarterly lags. The sample is reduced down to 116 to accommodate the testing of lags. The R2 statistic for the model is 0.953709 but the lags are all individually insignificant. However before removing them I will run an F-test to test their combined significance. The critical value for this F-test at 5% significance is , with

H0: β11 = β12 = β13 = β14 = 0

H1: Not all equal to zero.

The test statistic is computed as F = 0.6170 (4 d.p.) < 2.45. Therefore the null hypothesis is accepted and the lagged variables are removed. From the diagnostic tests it can be seen that the demand for wine shows insignificant signs of time lags at the 5% level of significance.

## Homogeneity and Slutsky

Demand Homogeneity

Demand homogeneity can be ‘referred to as the absence of money illusion’ (Chakravarty, 2005). If all prices double, then the quantity demanded will double according to microeconomic demand theory. Demand functions are in theory, homogenous of degree zero. To test for demand homogeneity, the following hypotheses must be drawn:

H0: 1 + 2 + 3 + 5+ 6 + 8 + 10 = 0

H1: 1 + 2 + 3 + 5+ 6 + 8 + 10 ≠ 0

The critical value is

Assuming H0, the model can be rewritten:

10 = -1 -2 -3 -5 -6 -8

The value for R2 is 0.952615 for the rewritten model when a regression was run.

The test statistic was computed as 0.01444 (4 s.f.) < 3.92. Therefore the null hypothesis is accepted and so demand homogeneity holds. The consumer does not suffer from money illusion in this case. The method used has come under scrutiny by Laitinen who claims that it is biased towards rejecting the null hypothesis (Laitinen, 1978). The data for wine (6) can be seen to be very strongly homogenous given this bias.

Slutsky

The Slutsky ‘equation portrays the income and substitution effects ... that a change in price will have on demand’ (Schotter, 2009). The Slutsky equation reads as follows:

¦U = constant

I will now manipulate the equation by multiplying through by :

¦U = constant

Which is equivalent to:

Price elasticity of demand = Substitution elasticity – Income elasticity x Proportion of income spent.

The income effect is found through multiplying the income elasticity by the proportion of income spent on wine. The substitution effect can be found by subtracting the income effect from the price elasticity of demand. From the brief, we are told that wine constitutes 4% of total consumption. The income elasticity can be found from table 6 to be 1.52185 and the price elasticity of demand can be found as -0.796731. Therefore:

Income effect = β10 x 0.04 = 1.52185 x 0.04 = 0.060874

Substitution effect = β6 – 0.060874 = – 0.796731 – 0.060874 = –0.857605.

As the income effect is positive, it shows that wine is a normal good. Due to marginal returns to scale, the substitution effect is always negative.

## Interpreting the preferred model

The preferred model can be found below with the standard errors in brackets:

LQWINE = 2.31584 – 1.25116LPMTFH – 0.893851LPFTVG – 0.459005LPTEA + 2.51069LPBEER –

(1.769) (0.2251) (0.2677) (0.2082) (0.183)

0.796731LPWINE – 0.706289LPTRAV + 1.52185LINCOME

(0.3533) (0.1736) (0.2539)

I will run the same diagnostic tests as in table 3 assuming the same hypotheses as in table 2.

Table 10

## Diagnostic Test

## Critical Value

## (5% significance)

## Preferred Model

Ramsey's RESET

3.92

3.8568

Jarque-Bera

5.99147

1.1456

F-test

2.18

321.7

White’s Test

1.64

1.0156

Breusch-Godfrey

2.29

0.56599

Durbin-Watson

dL = 1.462

dU = 1.898

2.14

R²

n/a

0.952621

xxxxxx - Pass, xxxxxx - Fail, xxxxxx - Indecision

The preferred model passes all of the diagnostic tests, except the Durbin-Watson test for which it lies in the region of indecision. It shows insignificant signs of abnormality, heteroskedasticity, auto-correlation and misspecification at the 5% level of significance. However, the model is closer to failing Ramsey’s Reset test than it was in its original form. Its F-test value is significantly higher than in the original model though. From this data I would say it is unnecessary to include any combinations of variables as all of the diagnostic tests are passed.

Having identified the preferred model, I will now look at the elasticities of the independent variables. As the function is double logarithmic, the elasticity of QWINE with respect to a variable is given by the coefficient of the log of that variable.

Table 11

## Parameter

## Coefficient

## Relationship with Wine (6)

Constant

2.31584

Autonomous

LPMTFH

-1.25116

Complementary

LPFTVG

-0.893851

Complementary

LPTEA

-0.459005

Complementary

LPBEER

2.51069

Substitutable

LPWINE

-0.796731

Wine (6) is price inelastic

LPTRAV

-0.706289

Complementary

LINCOME

1.52185

Wine (6) is income elastic

As was expected, wine is income elastic with a coefficient of 1.52185. This confirms its status as a luxury good. The coefficient for LPWINE is -0.796731 which suggests that wine is a popular household good and demand is not affected too greatly by price changes. This price elasticity may indicate habitual or addictive consumer behaviour. Meat and Fish, Fruit and Vegetables, Tea and Travel are all complementary goods and Beer is predictably a strong substitute, which is consistent with Pan, Fang and Malaga’s study.

## Conclusion

I have estimated a demand curve for wine (6) in Ruritania through using a series of diagnostic tests and analysis. The functional form was found to be double logarithmic. I then found that the prices of meat and fish, fruit and vegetables, tea, beer, wine, travel and the consumer income level are all significant influences on the quantity of wine demanded. The demand for wine showed no signs of a structural break, seasonality or time lags between 1980 and 2009. The demand for wine was found to be homogenous of degree zero and it has a positive income effect. This positive income effect when combined with a strong income elasticity of demand; strengthens wine’s status as a luxury good. Wine has an inelastic price elasticity of demand which may suggest it is being consumed on a partly addictive or habitual basis. Beyond this report, using variables such as the price of cheese or grapes for example would be useful in finding a more accurate demand curve for wine in Ruritania. A study by Barber, Taylor and Strick reveals the impact variables such as gender, location and age have on wine preferences. The ‘millennial generation’ makes its consumption choices with more regard for the environment than the ‘baby boomer’ generation (Barber, Taylor, & Strick, 2010). Including variables such as those studied by Barber, Taylor and Strick may provide insightful results.

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