Using the data provided, estimate an appropriate demand equation for fruit and vegetables and interpret the results.
Introduction
In this essay I will derive an adequate demand function for Fruit & Vegetables in Ruritania based on the quarterly time-series data assigned to me from the years 1987-2016. The data is for 10 variables that may affect levels of demand over a period of 30 years. First, I will identify each functional form the model could take and run diagnostic tests in order to select the most appropriate. I will then run further tests to determine any improvements that can be made to the model, after which I will test for demand homogeneity and other factors such as seasonality and structural breaks. Lastly, using the final refined model I will interpret the results, comparing my findings to any observations made in relevant literature.
Initial Predictions
Independent variable Predicted Effect
Price of meat and fish (PMTFH) Negative (strong). Meat and fish are complementary to vegetables so if price of meat and fish increases, demand for fruit and veg will likely fall.
Price of fruit and vegetables (PFTVG) Negative (weak). Fruit and vegetables are mostly price inelastic thus a rise in price of fruit and vegetables should not affect demand too much.
Price of tea (PTEA) Negative (weak). Tea may be complementary to fruit and vegetables, being served in similar situations.
Price of coffee (PCOFF) Same as above.
Price of beer (PBEER) No effect. Beer is not a complementary or substitute good for fruit and vegetables.
Price of wine (PWINE) Same as above
Price of leisure (PLEIS) Negative (weak). Whilst eating will take place in leisure time, it is unlikely that consumers will demand less fruit or vegetables should the price of leisure rise
Price of travel (PTRAV) Negative effect. Consumers need to travel to the shops to buy fruit and veg. If the price of travel rises then demand for fruit and vegetables may be negatively impacted.
Index of prices of all other goods (PALLOTH) Negative. Rising price levels will result in a fall in consumers’ disposable incomes and thus demand for any normal good would fall.
Income (INCOME) Positive. I would consider fruit and vegetables to be normal goods; demand should rise as income does.
Identifying functional forms
In order to find the most appropriate and fitting model for the data, I will test six different functional forms. These are:
Linear
Double log (log-log)
Semilog (log-lin)
Semilog (lin-log)
Inverse
Log-Inverse
The functional form that is chosen will need to satisfy all of the Gauss-Markov conditions for the ‘Best Linear Unbiased Estimator’ (BLUE) of the data. These assumptions are listed below.
The regression model is linear
The error term has a zero population mean
All explanatory variables are uncorrelated with the error term
Observations of the error term are uncorrelated with each other, (no autocorrelation)
The error term has a constant variance (homoscedasticity)
No explanatory variable is a perfect linear function of any other explanatory variable(s) (no multicollinearity)
The error term is normally distributed
I will test for these aspects after regressions are run on each model. The tests and corresponding hypotheses are outlined in the table below for a 5% significance level.
Diagnostic Test Tests for Hypotheses Critical values at 5% significance level
F-Test Overall significance H0: β1= β2 = … = βk where i = 1, 2, …, k. (Regressors are jointly significant)
H1: At least one βi ≠ 0. (Regressors are jointly insignificant F(10,109) ≈ 1.92
Durbin Watson First-order autocorrelation H0: ρ=0 (no autocorrelation)
H1A: ρ>0 (positive autocorrelation)
H1B: ρ<0 (negative autocorrelation)
4-dU = 2.11
4-dL = 2.47
dU = 1.89
dL = 1.53
Breusch-Godfrey Autocorrelation H0:ρ1=ρ2=0 (No autocorrelation)
H1:At least one ρi≠0 (Autocorrelation)
F(4,105) ≈ 2.46
Jarque-Bera Normality H0: Normality in error terms
H1: Error terms are not normally distributed χ2(2) ≈ 5.99
White (with no cross products) Heteroscedasticity H0: There is homoscedasticity
H1: There is heteroscedasticity F(20,99) ≈ 1.68
White (with cross products) Heteroscedasticity H0: There is homoscedasticity
H1: There is heteroscedasticity F(65, 54) ≈ 1.54
Ramsey RESET Model misspecification H0: γ1=0 (Model has correct functional form and all necessary variables)
H1: γ1≠0 (Model is mis-specified)
F(1,108) ≈ 3.93
Estimated regression equations:
Linear:
QFTVG=585466-602904PMTFH+167023PFTVG-35793.8PTEA+14966.3PCOFF+195631PBEER-46171PWINE+1847.12PLEIS+2307.28PTRAV-211.099PALLOTH-216.707INCOME
Double Log:
lnQFTVG=8.20-0.642lnPMTFH-0.810lnPFTVG-0.364lnPTEA+0.290lnPCOFF+0.112lnPBEER-0.397lnPWINE+0.135lnPLEIS+0.323lnPTRAV+1.57lnPALLOTH-0.895lnINCOME
Log-Lin:
lnQFTVG=11.996-0.501PMTFH-2.12PFTVG-0.263PTEA+0.0368PCOFF+0.0108PBEER+0.0210PWINE+0.000911PLEIS+0.00437PTRAV+0.00938PALLOTH-0.000370INCOME
Lin-Log:
QFTVG=(-1.291e+06)-807241lnPMTFH-121249lnPFTVG-314204lnPTEA-191618lnPCOFF+618037lnPBEER-862300lnPWINE+422452lnPLEIS+114542lnPTRAV+133096lnPALLOTH-68505.6lnINCOME
Inverse:
QFTVG=-885821+659114 1/PMTFH+71893.4 1/PFTVG+582004 1/PTEA+(1.59e+06) 1/PCOFF-940782 1/PBEER+(6.76e+06) 1/PWINE+300975 1/PTRAV-(1.04e+08) 1/PALLOTH-(8.19e+08)1/INCOME
Log-Inverse:
lnQFTVG=11.6+0.503 1/PMTFH+0.224 1/PFTVG+0.566 1/PTEA-2.23 1/PCOFF-0.118 1/PBEER+3.91 1/PWINE+15.2 1/PLEIS-6.21 1/PTRAV-354.3 1/PALLOTH+2016.12 1/INCOME
Functional form
R2 RSS F-Test DW BG (lags 1-4) JB White White (cross products) RR
Linear 0.805 1.45e+13 45.1* 1.93 0.128 73.5* 3.74* 7.97* 37.0*
Double Log 0.931 16.6 148* 1.86 1.85 1.73 1.18 1.26 0.588
Log-Lin 0.933 16.3 151* 1.85 1.63 1.22 0.945 1.10 0.128
Lin-Log 0.781 1.64e+13 38.9* 1.82 1.20 125* 2.17* 2.71* 32.9*
Inverse 0.773 1.69e+13 37.2* 1.80 1.88 134* 1.29 1.39 23.3*
Log-Inverse 0.926 18.0 136* 1.83 1.90 3.97 1.36 1.35 0.122
*Significance at a 5% level i.e. the null hypothesis can be rejected.
The Linear, Lin-Log and Inverse models do not pass all of the tests for the Gauss-Markov assumptions and thus they can be ruled out.
Whilst all of the R2 values for the remaining models are very similar, I will rule out the Log-Inverse model as this has the lowest value of 0.926. For the remaining two models, there is not much difference in the results of the regression and thus no particular reason for any preference over the two models. Thus, I have chosen my preferred functional form based on the ease with which the results can be interpreted. The Double Log model does seem to be more appropriate in this sense. This functional form was also adopted by Deaton and Muellbauer (1980) in their Almost Ideal Demand System (AIDS), upon which many demand models in the literature are based, lending it further credibility as a suitable functional form.
Deriving the preferred model
I will now carry out t-tests to test the individual significance of the variable coefficients. These will be two tailed tests at the 5% significance level.
lnQFTVG=8.20-0.642lnPMTFH-0.810lnPFTVG-0.364lnPTEA+0.290lnPCOFF+0.112lnPBEER-0.397lnPWINE+0.135lnPLEIS+0.323lnPTRAV+1.57lnPALLOTH-0.895lnINCOME
H_0:β_i=0 (Variable is insignificant)
H_1:β_i≠0 (Variable is significant)
The null hypothesis will be rejected if the test statistic is greater (less) than the positive (negative) critical value (t_(2.5%,108)=±1.98).
The model does pass all of the diagnostic tests. However, as shown by the results, there are five independent variables that prove to be insignificant to the model: lnPTEA, lnPCOFF, lnPBEER, lnPWINE and lnPLEIS. I initially anticipated that these variables would likely have a weak or no effect on demand for fruit and vegetables so it follows that they would be insignificant when tested. These regressors can be taken out of the model, however an F-test for joint significance should be carried out to ensure that the new model would be an improvement of the original.
Variable βi
Standard error t-stat Significance at 5% t-prob
Constant 8.20 2.70 3.03 Significant 0.00
lnPMTFH -0.64 0.18 -3.53 Significant 0.00
lnPFTVG -0.81 0.29 -2.76 Significant 0.01
lnPTEA -0.36 0.26 -1.38 Insignificant 0.17
lnPCOFF 0.29 0.27 1.08 Insignificant 0.28
lnPBEER 0.11 0.24 0.47 Insignificant 0.64
lnPWINE -0.40 0.32 -1.23 Insignificant 0.22
lnPLEIS 0.14 0.18 0.77 Insignificant 0.45
lnPTRAV 0.32 0.16 2.07 Significant 0.04
lnPALLOTH 1.57 0.54 2.90 Significant 0.00
lnINCOME -0.90 0.22 -4.09 Significant 0.00
Unrestricted model:
lnQFTVG=8.20-0.642lnPMTFH-0.810lnPFTVG-0.364lnPTEA+0.290lnPCOFF+0.112lnPBEER-0.397lnPWINE+0.135lnPLEIS+0.323lnPTRAV+1.57lnPALLOTH-0.895lnINCOME
Restricted model 1:
lnQFTVG=5.85-0.522lnPMTFH-0.809lnPFTVG+0.258lnPTRAV+2.46lnPALLOTH-1.07lnINCOME
H_0: β_3=β_4=β_5=β_6=β_7=0 (Jointly insignificant)
H_1: At least one β_i≠0 where i=3,4,5,6 or 7
F-statistic=(〖(RSS〗_R-〖RSS〗_U)/M)/(〖RSS〗_U/(N-K-1))
M is the number of restrictions, K is the number of variables estimated in the unrestricted model; 〖RSS〗_R concerns the restricted model, and 〖RSS〗_U the unrestricted model.
F-statistic=((18.4707-16.642)/5)/(16.642/(120-10-1))=2.40
The F critical value is found by F(5,109) = 2.30. 2.40 > 2.30 hence we must reject H_0, showing that at least one of the excluded variables is significant.
Of all of the excluded variables, lnPTEA seemed to be the least insignificant when the t-tests were run. For this reason, I will carry out the F-test again, with the inclusion of lnPTEA in the restricted model. Whilst statistically this variable is still insignificant to the overall demand function, I have opted to leave in it the model; it is not irrational to suggest that changes in the price of tea could affect demand for fruit and vegetables. As mentioned before, tea is often consumed at similar times to fruit, at teatime for example, or as part of a snack.
H_0: β_4=β_5=β_6=β_7=0 (Jointly insignificant)
H_1: At least one of β_i≠0 where i=4,5,6 or 7
RSSR = 17.330
F statistic = 1.13
F-critical value = F(4,109) = 2.45
1.13 < 2.45, thus with the inclusion of lnPTEA, we fail to reject the null hypothesis, proving that at the 5% significance level, the excluded variables are jointly insignificant. This allows us to impose the following linear restriction on the model.
Restricted model 2:
lnQFTVG=6.04-0.471lnPMTFH-0.984lnPFTVG-0.472lnPTEA+0.167lnPTRAV+2.21lnPALLOTH-0.884lnINCOME
As shown by the results of the regression below, the new restricted model also passes all diagnostic tests. Therefore, we are able to use this as the final preferred model.
Calculated value Critical value Significance at 5%
F-test 245 F(6,113) = 2.18 Significant
DW 1.80 K=7 dL=1.62, dU=1.79 Insignificant
BG (1-4) 1.13 F(4,109) = 2.45 Insignificant
JB 1.72 χ22=5.99
Insignificant
White 0.929 F(12,107) = 1.84 Insignificant
White X 1.13 F(27,92) = 1.61 Insignificant
RR 3.36 F(1,112) = 3.93 Insignificant
Demand Homogeneity
If both prices and income rise by the same amount, homogeneity means that the level of demand should be unaffected.
If demand homogeneity exists, β1 + β2 + β3 + β8 + β9 + β10 = 0. Running an F-test on the following regression allows us to test for this.
β10 = – β1 – β2 – β3 – β8 – β9.
lnQFTVG=β_0+β_1 lnPMTFH+β_2 lnPFTVG+β_3 lnPTEA+β_8 lnPTRAV+β_9 lnPALLOTH-(β_1+β_2+β_3+β_8+β_9)lnINCOME
=β_0+β_1 (lnPMTFH-lnINCOME)+β_2 (lnPFTVG-lnINCOME)+β_3 (lnPTEA-lnINCOME)+β_8 (lnPTRAV-lnINCOME)+β_9 (lnPALLOTH-lnINCOME)
lnQFTVG=β_0+β_1 ln PMTFH/INCOME+β_2 ln PFTVG/INCOME+β_3 ln PTEA/INCOME+β_8 ln PTRAV/INCOME+β_9 ln PALLOTH/INCOME
H_0:β_1+β_2+β_3+β_8+β_9+β_10=0 (Demand is homogeneous of degree 0)
H_1:β_1+β_2+β_3+β_8+β_9+β_10≠0 (Demand is not homogeneous of degree 0)
RSSR = 17.410
RSSU = 17.330
F-critical value = F(1,113) = 3.93
F-statistic=(〖(RSS〗_R-〖RSS〗_U)/M)/(〖RSS〗_U/(N-K-1))
=((17.410-17.330)/1)/(17.330/(120-6-1))=0.52
F statistic < F critical value; we fail to reject H0, indicating that the model does indeed exhibit demand homogeneity of degree 0.
Slutsky equation
The Slutsky equation will examine the effects of a change in the price of fruit and vegetables on consumers, more specifically the magnitude and direction of any income or substitution effects. The equation is:
∂QFTVG/∂PFTVG=├ ∂QFTVG/∂PFTVG┤|_(u=constant)-∂QFTVG/∂INCOME*QFTVG
or:
e_(x,p_x )=e_(x,p_x)^s-s_x e_(x,I)
e_(x,p_x)^s=the elasticity of substitution for fruit and vegetables
s_x=the proportion of income spent on fruit and vegetables,in Ruritania this is 6%
e_(x,p_x )=the price elasticity of demand
e_(x,I)=the income elasticity of demand
The price elasticity of demand is estimated to be -0.984, and the income elasticity of demand is -0.884.
e_(x,p_x)^s=e_(x,p_x )+s_x e_(x,I)=(-0.984)+0.06(-0.884)=-1.03
Substitution effect = -1.03
Income effect = 0.06*(-0.884) = -0.0530
The results show that demand for fruit and vegetables is price inelastic as |e_(x,p_x ) |<1. Moreover, the substitution effect is negative, meaning that consumers of Fruit & Vegetables in Ruritania would happily substitute 1.03% of their Fruit & Vegetable consumption should there be a 1% rise in the price of this good. The negative income effect however suggests Fruit & Veg in Ruritania is not a normal good but in fact inferior, implying that as income rises, demand for the good falls. The results show that if there is such a price rise of 1% in Ruritania, the fall in relative purchasing power of consumers will actually result in an increase in consumption of Fruit & Veg of 0.053%. Whilst unexpected to what I initially predicted, it follows that rather than filling meals with Fruit & Vegetables, a rise in income could mean that consumers are able to substitute towards meat and fish products, which are on average more expensive.
Additional tests
1. Structural stability
A structural change in the data will mean that one regression model is insufficient in explaining the data. I have thus decided to run a Chow test for structural stability.
To aid in this test, I have graphed demand against time to see how levels have changed over time. This will determine the time periods I will use.
From the graph above we can see that there is particularly rapid growth in demand after the year 2003, before which levels of demand rise slowly and gradually. Thus I will set the two periods to 1987(1) to 2002(4) and 2003(1) to 2016(4).
H_0: γ_i=α_i,i= 0, 1, 2, 3, 8, 9, 10(No structural change)
H_1: At least one γ_i≠α_i (Structural change)
Period 1(α): 1987(1) – 2002(4) RSS1 = 10.2845 n1 = 64
Period 2(γ): 2003(1) – 2016(4) RSS2 = 5.9444 n2 = 56
F-statistic=(〖(RSS〗_N-〖RSS〗_U)/K)/(〖RSS〗_U/(N-2K))
K is the number of coefficients estimated = the number of restrictions imposed = 7
N = n1 + n2 = 64 + 56 = 120
RSSN is the pooled restricted model = 17.330
RSSU = RSS1 + RSS2 = 10.2845 + 5.9444 = 16.229
F-statistic = ((17.330-16.2289)/7)/(16.2289/(106)) = 1.03
F-critical value at 5% significance level is given by F(K, N-2K) = F(7, 106) = 2.10
As the F-statistic < F-critical value, at the 5% level, there is insignificant evidence to suggest that there is structural instability in the data; we cannot reject the null hypothesis. We are able to keep the original restricted pooled model as the most appropriate function for demand for Fruit & Vegetables in Ruritania.
2. Seasonality
Like most food products, demand for Fruit & Vegetables may change seasonally. This is particularly true for this good as various Fruit & Vegetable products can only be grown during certain times of year, which could in turn contribute to overall demand levels. The test I will carry out will determine whether there is indeed a significant seasonality effect.
Unrestricted model:
lnQFTVG=β_0+β_1 lnPMTFH+β_2 lnPFTVG+β_3 lnPTEA+β_8 lnPTRAV+β_9 lnPALLOTH+β_10 lnINCOME+β_11 S1+β_12 S2+β_13 S3
Si=1 in Season i, and 0 otherwise, i = 1,2 or 3
H_0: β_11=β_12=β_13=0 (No seasonality)
H_1: At least one β_i≠0 where i=11,12 or 13 (There is a seasonality effect)
F-statistic=(〖(RSS〗_R-〖RSS〗_U)/M)/(〖RSS〗_U/(N-K-1))
RSSU = 16.687
RSSR = 17.330
M = 3, K = 9
F-statistic = 1.41
F-critical value at the 5% significance level = F(3, 110) = 2.69
The F-statistic < F-critical value; we must accept the null hypothesis, there is insignificant evidence to suggest that there are seasonal changes in demand for Fruit & Vegetables. One reason for this could be that any reductions in demand for a certain fruit or vegetable in one season may be offset by a rise in demand of another good, resulting in no overall change to demand. Another explanation could be the availability of all fruits and vegetables year round in Ruritania.
3. Lagged variables
Lastly, I will add lagged variables for all four quarters to the model to test whether the price in previous quarters will affect the current consumption of Fruit & Vegetables in Ruritania. I will test for lags in the PFTVG variable, carrying out a t-test for the individual significance of each lagged dependent variable.
H_0:β_i=0 (Variable is insignificant)
H_1:β_i≠0 (Variable is significant)
The null hypothesis will be rejected if the test statistic is greater (less) than the positive (negative) critical value. The critical value is t_(2.5%,108)=±1.984.
Variable βi
t-stat Significance at 5% t-prob
Constant 6.12 3.06 Significant 0.00
lnPMTFH -0.54 -3.71 Significant 0.00
lnPFTVG -0.93 -2.09 Significant 0.04
lnPFTVG_1 0.08 0.13 Insignificant 0.90
lnPFTVG_2 0.24 0.38 Insignificant 0.71
lnPFTVG_3 -0.64 -1.00 Insignificant 0.32
lnPFTVG_4 0.26 0.55 Insignificant 0.59
lnPTEA -0.48 -2.62 Significant 0.01
lnPTRAV 0.23 1.69 Insignificant 0.10
lnPALLOTH 2.12 6.01 Significant 0.00
lnINCOME -0.88 -4.29 Significant 0.00
The lagged terms all seem to be insignificant at a 5% significant level suggesting no lagged effects of the PFTVG variable.
I also carried out an F-test for joint significance, which yielded the same conclusion – the lagged variables have no effect on current levels of demand and will not be included in the model.
Presentation and Interpretation of the Preferred Model
I am able to conclude that the demand for Fruit & Vegetables in Ruritania can be accurately modelled by the following Double Log equation.
lnQFTVG=6.04-0.471lnPMTFH-0.984lnPFTVG-0.472lnPTEA
(1.72) (0.140) (0.248) (0.173)
+0.167lnPTRAV+2.21lnPALLOTH-0.884lnINCOME
(0.132) (0.280) (0.199)
R2 = 0.929 RSS = 17.3 n = 120
The model passes all diagnostic tests at the 5% significance level – further verified by the graphs below that show the existence of normality and homogeneity in the error terms. It can also be noted that the model is able to explain 92.9% of the data.
The coefficients in this model represent the elasticities of demand for the corresponding variables.
Price elasticity of demand was estimated to be -0.984 meaning that if there is a 1% price rise of Fruit & Vegetables there will be a 0.984% fall in demand. This value is indeed inelastic, however it is very close to unit elasticity.
The cross price elasticity of demand of Meat & Fish is -0.471, the negative value implying that Meat & Fish is complementary to Fruit & Vegetables. This effect is lower than I anticipated, with the price of Meat & Fish having a weak effect on demand for Fruit & Vegetables. A similar value of – 0.472 was observed for the cross price elasticity of Tea, showing that Tea and Fruit & Vegetables are complementary goods. A 1% price rise in Tea would result in a fall in demand for Fruit & Veg by 0.472%.
An unusual cross price elasticity result is that of the price of Travel. The value of 0.167 suggests that the two variables are substitutes, although this is weak. One reason for this may be that consumers may substitute towards to foods such as Fruit & Vegetables that may already be available at home, rather than travelling to retail outlets to purchase other foods.
The cross price elasticity of All Other Goods to Fruit & Veg in Ruritania is 2.21, indicating a strong positive effect of a rise in price of all other goods. Clearly, consumers in Ruritania are likely to substitute other goods for Fruit & Vegetables if prices of other goods rise.
Lastly, the income elasticity of demand is -0.884. As mentioned before, this means that Fruit & Veg is an inferior good. It is important to note that individual income elasticities of various fruits and vegetables may indeed be positive, however the overall effect implies an inferior relationship. A 1% rise in national income would result in a 0.884% fall in demand for Fruit & Vegetables. Huang (1985) breaks down the income elasticities and shows that the in the US there are indeed a number of fruits and vegetables that are inferior goods, suggesting that the result here is not unusual.
Fruit and vegetables play an essential part in most diets; they have well-known health benefits and are consumed globally. In fact, many governments do advocate for regular consumption (for example, the UK’s 7-a-day scheme). In much of the literature, Fruit & Vegetables are recognised as price inelastic goods, De Agostini (2014) estimated that a price elasticity value for the UK is -0.619 and notes that the value has become more elastic over time. One reason for increased elasticity could be the increasing availability of other sources of nutrition due to the popular trend of healthy living and veganism that has been seen globally in recent years. Additionally, Huang and Lin (2000) observed similar values for fruit and vegetable demand in the US in the range of -0.72 and -1.01, which is also consistent with the results from Ruritania. Whilst in these cases, demand is more price inelastic, there is still a negative effect of prices on demand. With regards to cross price elasticities, there is much variation in the values suggesting that the effect of prices of other goods is very much dependent on consumer preferences and trends in these countries.
To conclude, my analysis of the data does overall seem consistent with the findings for various countries in the relevant literature. The use of the Double Log model is supported by the AIDS model by Deaton and Muellbauer and the subsequent studies of demand in later papers. Moreover, both data from the US and UK also support the elasticity results, suggesting that, as an estimation of demand levels for fruit and vegetables in Ruritania, the model is reasonably accurate. Nonetheless there are a number of limitations that, if accounted for, could certainly improve the model. In the majority of the literature, demographic categories are included in demand models, with authors examining the effects of age, gender and different income households on levels of demand for fruit and vegetables. In addition, as Huang (1985) demonstrates, there are likely to be very different elasticities for the various fruits and vegetables, which could have a distortionary effect on the overall values. For a more efficient model, these factors could perhaps be incorporated. However, for the information provided to me, I believe I have estimated the best model for demand of fruit and vegetables in Ruritania.
Word count: 2468
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