**Preamble**

The law of demand states that demand of a given commodity is inversely related to its market price holding other factors constant (Cameron & Trivedi 2014). However, demand of a commodity is directly related to the market price of a substitute product and the amount of consumer disposable income (Ghylsels & Marcellino 2016). This paper seeks to estimate the demand function for a non-durable consumer goods [clothing and shoes(CLOT)] and a durable commodity [furniture (FURN)] using annual U.S. time series data related to the period, 1959-2003. A log linear regression model will be estimated for CLOT and FURN. The paper will rely on TSM’s statistical output results to test for the law of demand, homogeneity restriction hypothesis and perform the parameter stability testing.

**Methods**

The paper will employ Ordinary Least Squares (OLS) regression method to estimate the demand equation for CLOT and FURN. The estimated OLS model for CLOT and FURN will be used to forecast demand for year 2003 based on the actual price and disposable income data. In addition, t-test and TSM’s Wald test will be used to perform statistical homogeneity restriction tests for the stated time series data. Finally, parameter stability for the stated time series data will be tested using Chow test.

**Results and Description**

**Basic Demand Model**

*1. Estimation of Demand Equation for CLOT and FURN*

Log-linear models of the following form will be estimated based on the OLS results from TSM; Log CLOT = β_{0} + β_{p}Log PCLOT + β_{q}Log QCLOT + β_{y}Log DPI and Log FURN = β_{0} + β_{p}Log PFURN + β_{q}Log QFURN + β_{y}Log DPI. Table 1 and 2 present a summary of the OLS regression analysis results to estimate the demand function for CLOT and FURN.

**Table 1: Summary of OLS Regression Analysis (CLOT)**

**Table 2: Summary of OLS Regression Analysis (FURN)**

The following models are estimated based on results of OLS regression analysis for both CLOT and FURN.

Log CLOT = -1.14 – 0.74Log PCLOT + 0.66Log QCLOT + 0.81Log DPI

Log FURN = -4.10 – 1.14Log PFURN + 0.75Log QFURN + 1.13Log DPI

*Confidence Interval for Income Elasticities (CLOT)*

Upper t-critical value (t_{0.05}) = 2.01954

P [0.80804 – (0.03802*2.01954)] <ε_{y}< [0.80804 + (0.03802*2.01954)] = 95%

P (0.7313 <ε_{y}< 0.8848) = 95%

*Confidence Interval for Income Elasticities (FURN)*

Upper t-critical value (t_{0.05}) = 2.01954

P [1.1336 – (0.07597*2.01954)] <ε_{y}< [1.1336 + (0.07597*2.01954)] = 95%

P (0.9802 <ε_{y}< 1.2870) = 95%

The results based on CLOT indicate that there is a 95% likelihood of the consumers’ income elasticity lying between 0.7313 and 0.8848. The implication is that consumers’ demand for clothing and shoes (CLOT) is highly inelastic to changes in personal disposable income.

The results for FURN illustrate that there is a 95% chance of the consumers’ income elasticity ranging between 0.9802 and 1.2870. The implication is that based on FURN, consumers’ demand for furniture products is highly elastic to fluctuations in disposable income. Therefore, a comparison of the results indicates that while demand for CLOT is highly inelastic to changes in income, demand for FURN seems to be relatively elastic to fluctuations in personal disposable income.

*Confidence Interval for Own Price Elasticities (CLOT)*

Upper t-critical value (t_{0.05}) = 2.01954

P [-0.7365 – (0.03907*2.01954)] <ε_{y}< [-0.7365 + (0.03907*2.01954)] = 95%

P (-0.8154 <ε_{p}< -0.6576) = 95%

*Confidence Interval for Own Price Elasticities (FURN)*

Upper t-critical value (t_{0.05}) = 2.01954

P [-1.13706 – (0.12723*2.01954)] <ε_{y}< [-1.13706 + (0.12723*2.01954)] = 95%

P (-1.3940 <ε_{p}< -0.8801) = 95%

The results based on CLOT depict that there is a 95% likelihood of the consumers’ own price elasticity lying between -0.8154 and -0.6576. The implication is that consumers’ demand for clothing and shoes is relatively inelastic to changes in price for CLOT. This is because, the stated own price elasticity range is less than 1.

The outcome based on FURN depict that there is a 95% likelihood of the consumers’ own price elasticity ranging between -1.3940 and -0.8801. The implication is that in comparison to CLOT, consumers’ demand for furniture products is relatively elastic to changes in price. This is because, own price elasticity of FURN is slightly closer to 1 and/or greater than 1.

*2. Testing the Law of Demand based on the Demand Function for CLOT and FURN*

The following are the null and alternative hypothesis that will be used in testing the law of demand based on the demand function for CLOT and FURN.

H_{o}: β_{p} = 0

H_{a}: β_{p} < 0

The results based on OLS regression analysis for CLOT indicate that t statistic abs (t = -18.851) is greater than t-critical (t = 2.01954). The implication is that the analysis does not accept the null hypothesis, which postulates that price has no effect on demand for CLOT and concludes that the law of demand holds.

In the case of FURN, the results based on OLS regression analysis depict that t-statistic abs (t = -8.937) is greater than t-critical (t = 2.01954). Similarly, the analysis does not accept the null hypothesis, which postulates that price has no effect on demand for furniture products (FURN) and alternatively concludes that the law of demand holds.

This means that at 5% significance level, there is significant statistical evidence to suggest that the law of demand holds in the time series data for both CLOT and FURN, which illustrate that there is a negative relationship between price and demand for clothing and furniture products in the U.S. In testing the law of demand, Talukder (2001) found that there is a significant negative relationship between price and quantity demanded for commodities.

*3. Testing the Homogeneity Restriction*

*3(a) Testing Homogeneity Restriction using t-test*

The following null and alternative hypothesis will be used to perform test for homogeneity restriction based on the demand function for CLOT and FURN.

H_{o}: β_{p }+ β_{q} = 0

H_{a}: β_{p }+ β_{q} ≠ 0

Table 3 and 4 present a summary of the OLS linear regression results after replacing Log PCLOT and Log PFURN with Log PREL in the respective models.

**Table 3: Summary of OLS Linear Regression Analysis with Log PREL (CLOT)**

**Table 4: Summary of OLS Linear Regression Analysis with Log PREL (FURN)**

Given that ρ-value for Log QCLOT is 0.076 > 0.05, the analysis indicates that the estimated coefficient for Log QCLOT (β_{q} = -0.07633) is not statistically significant. In addition, t-statistic abs (t = -1.819) is less than t-critical (t = 2.01954). The implication is that the analysis accepts the null hypothesis and concludes that the homogeneity restriction condition holds in the demand function for Log CLOT at 5% significance level.

However, ρ-value for Log QFURN is 0.000 < 0.05, which implies that the estimated coefficient for Log QFURN (β_{q} = -0.39159) is statistically significant. Furthermore, t-statistic abs (t = -5.905) is greater than t-critical (t = 2.01954). The implication in the case of Log QFURN is that the analysis does not accept the null hypothesis and concludes that the homogeneity restriction condition fails to hold in the demand function for Log FURN at 5% significance level. Therefore, while the homogeneity restriction condition holds in the case of CLOT, it fails to meet the statistical threshold in the case of FURN.

*3(b) Testing Homogeneity Restriction using TSM’s Wald Test*

The following null and alternative hypothesis will be used to perform test for homogeneity restriction using TSM’s Wald Test.

H_{o}: β_{p }+ β_{q} = 0

H_{a}: β_{p }+ β_{q} ≠ 0

Table 5 and 6 present a summary of the OLS linear regression results and TSM’s Wald test outcome for both CLOT and FURN.

**Table 5: Summary of OLS Linear Regression and TSM’s Wald Test (CLOT)**

**Table 6: Summary of OLS Linear Regression and TSM’s Wald Test (FURN)**

Table 5 depicts that the Chi-square statistic is (χ^{2} = 3.3076) based on the outcome of Wald test for CLOT. Therefore, given that χ^{2} statistic (3.3076) < χ^{2}_{0.05 }(7.81), the analysis accepts the null hypothesis and concludes that the homogeneity restriction holds in the log linear demand function for CLOT.

TSM’s Wald test results for FURN depicted in table 6 indicate that the Chi-square statistic is (χ^{2} = 34.8779). Therefore, given that χ^{2} statistic (34.8779) > χ^{2}_{0.05 }(7.81), the analysis fails to accept the null hypothesis and concludes that the homogeneity restriction does not hold in the log linear demand function for FURN.

The stated two methods (t-test and Wald test) should generate the same outcome because both test the significance of the parameter estimates β_{p} (Log PCLOT and Log PFURN) and β_{q} (Log QCLOT and Log QFURN).

*4. Projected Demand for Year 2003 using the Estimated Model (1959-2002)*

Table 7 and 8 present a summary of the OLS linear regression analysis outcome based on the U.S. time series data related to the period, 1959-2002.

**Table 7: Summary of OLS Linear Regression Model (1959-2002) (CLOT)**

**Table 8: Summary of OLS Linear Regression Model (1959-2002) (FURN)**

The following OLS linear regression models for CLOT and FURN are estimated based on the stated U.S. time series data related to the period, 1959-2002.

Log CLOT = -1.16 – 0.77Log PCLOT + 0.66Log QCLOT + 0.82Log DPI

Log FURN = -4.08 – 1.21Log PFURN + 0.79Log QFURN + 1.15Log DPI

The actual commodity prices and income for the year 2003 are stated as follows;

Actual year 2003 price (PCLOT) = 93.045 and Log PCLOT = 1.969

Actual year 2003 price (PFURN) = 95.025 and Log PFURN = 1.978

Actual year 2003 income (DPI) = 7.733.8 and Log DPI = 3.888

Therefore, the demand forecast for clothing (CLOT) and furniture (FURN) related to year 2003 is estimated as follows;

*Clothing (CLOT)*

Log CLOT = -1.16 – (0.77*1.969) + (0.66*1.978) + (0.82*3.888) = 1.8175

Projected price of CLOT for year 2003 = 10^{1.8175 }= 65.69

Actual forecast error for PCLOT = 65.69-93.045 = -27.35

*Furniture (FURN)*

Log FURN = -4.08 – (1.21*1.978) + (0.79*1.969) + (1.15*3.888) = -0.44667

Projected price of FURN for year 2003 = 10^{-0.44667 }= 0.3575

Actual forecast error for PFURN = 0.3575-95.025 = -94.6675

*Chow Forecasting Test*

Chow forecasting tests estimates whether the regression coefficients are the same or different for a pair of split data points (Chang & McAleer 2015). The following null and alternative hypothesis will be used to examine whether data used in generating the forecast (1959-2002) is drawn from the same sample period, 1959-2003.

H_{o}: ε_{1} = ε_{2}

H_{a}: ε_{1} ≠ ε_{2}

Table 7 depicts that the Chi-square statistic is (χ^{2} = 5.1854) based on the outcome of Chow test. Given that χ^{2} statistic (5.1854) < χ^{2}_{0.05 }(7.81), the analysis accepts the null hypothesis and concludes that the error value from the time series data, 1959-2002 has the same mean and variance with the error value associated with the sample period, 1959-2003.

Similarly, table 8 depicts that the Chi-square statistic based on Chow test (χ^{2} = 2.5214) < χ^{2}_{0.05 }(7.81). The implication is that the analysis accepts the null hypothesis and concludes that similar to CLOT, the error value from the FURN time series data, 1959-2002 has the same mean and variance with the error value associated with the sample period, 1959-2003.

*5. Evaluating the Effect of Population (Log POP)*

Table 9 and 10 present a summary of the OLS linear regression analysis results for estimating the demand function based on Log CLOT and Log FURN.

**Table 9: Summary of OLS Linear Regression Model with Addition of Population (CLOT)**

**Table 10: Summary of OLS Linear Regression Model with Addition of Population (FURN)**

The results depicted in table 9 indicate that the coefficient of Log POP = -0.28157 with ρ-value (0.528) > 0.05. The implication is that the effect of population on the demand for clothing (CLOT) is not statistically significant at 5% significance level. Similarly, the results depicted in table 10 indicate that the coefficient of Log POP = -0.1458 with ρ-value (0.885) > 0.05. The results based on FURN seem to be consistent with findings of CLOT, which concur that the effect of population on demand is not statistically significant at 5% significance level.

If the coefficient of Log POP (β_{z}) = 1-β_{y}, the implication is that β_{z} + β_{y} = 1. Therefore, consumer expenditure and income data should be expressed as a per-capita in order to improve stability of the estimated model. This is because, when consumer expenditure and income data is expressed in aggregate rather than per-capita form, the stated variable is likely to fluctuate in response to changes in population. This is justified by the fact that as population changes, expenditure and income also changes in the same direction (Gujarati & Porter 2016; Greene 2017).

*Wald Test of Coded Restriction*

H_{o}: β_{z }= 1- β_{y}

H_{o}: β_{z }≠ 1- β_{y}

Where β_{z }= coefficient of Log POP.

Table 11 and 12 present a summary of the results based on Wald test of coded restriction to evaluate the significance of transforming expenditure and income from aggregate value to per-capita value.

**Table 11: Summary of the Results based on Wald Test of Coded Restriction (CLOT)**

**Table 12: Summary of the Results based on Wald Test of Coded Restriction (FURN)**

Table 11 depicts that the Chi-square statistic is (χ^{2} = 1.5515) based on the outcome of Wald test of coded restriction for CLOT. Given that χ^{2} statistic (1.5515) < χ^{2}_{0.05 }(7.81), the analysis accepts the null hypothesis and concludes that the coefficient of population (β_{z}) is equal to 1- β_{y. }Similarly, the results presented in table 12 indicate that the Chi-square statistic is (χ^{2} = 0.0013) based on the outcome of Wald test of coded restriction for FURN. Given that χ^{2} statistic (0.0013) < χ^{2}_{0.05 }(7.81), the analysis accepts the null hypothesis and concludes that the coefficient of population (β_{z}) is equal to 1- β_{y.}

The implication is that both sets of results agree consumer expenditure and income data should be transformed from aggregate value to per-capita in order to improve the model stability (Wooldridge 2016).

*7. Chow Parameter Stability Test*

Chow parameter stability test seeks to examine whether the coefficients would remain the same as the model’s structural attributes changes (Pedace 2013). This section of the paper will test whether coefficients of the estimated OLS model (1959-1980) are statistically different from the coefficients of the estimated OLS model (1981-2003) based on the following null and alternative hypothesis.

H_{o}: β_{1 }= β_{2}

H_{o}: β_{1 }≠ β_{2}

Table 13 and 14 present the results based on Chow parameter stability tests for both CLOT and FURN.

**Table 13: Summary of the Outcome based on Chow Parameter Stability Test (CLOT)**

**Table 14: Summary of the Outcome based on Chow Parameter Stability Test (FURN)**

Table 13 indicates that the Chi-square statistic is (χ^{2} = 38.3379) based on the outcome of Chow parameter stability test for CLOT. Given that χ^{2} statistic (38.3379) > χ^{2}_{0.05 }(7.81), the analysis fails to accept the null hypothesis and concludes that the parameter coefficients for the time series data (1959-1980) are statistically different from the parameter coefficients related to the ex-post time series data (1981-2003).

The results based on Chow parameter stability test for FURN presented in table 14 also seem to be consistent with the outcome for CLOT. This is because, the Chi-square statistic (χ^{2} = 72.5486) > χ^{2}_{0.05 }(7.81), which implies that the analysis fails to accept the null hypothesis and concludes that the parameter coefficients for the time series data (1959-1980) are statistically different from the coefficients associated with the ex-post time series data (1981-2003). The implication is that there is substantial change in the estimated model parameters due to structural changes in the model (Stock & Watson 2017; Studenmund 2017).

*8. Inclusion of the Trend Dummy Variables*

This section of the paper will examine the effect of introducing time variable (trend dummy) in the estimated OLS demand function for CLOT and FURN. Table 15 and 16 present a summary of the OLS linear regression results with inclusion of a trend dummy variable.

**Table 15: Summary of OLS Regression with Inclusion of Trend Dummy Variable (CLOT)**

**Table 16: Summary of OLS Regression with Inclusion of Trend Dummy Variable (FURN)**

In the case of CLOT, the ρ-value for Trend variable is (ρ = 0.27>0.05). The outcome for FURN also seems to be consistent with the results based on CLOT. This is because, the ρ-value for Trend variable (ρ = 0.321) is greater than 0.05. The implication is that for both CLOT and FURN, the time variable (Trend) is not statistically significant at 5% significance level. In addition, from the OLS linear regression results depicted in table 8, there is no substantial change in the coefficients for price and income variables with inclusion of trend compared to the previous tests and models.

**Conclusion**

This paper sought to test for validity of the law of demand, homogeneity restriction and the parameter stability using U.S. consumer time series data related to the period, 1959 to 2003. Using OLS regression analysis results, the paper finds and acknowledges that the law of demand holds for the estimated demand function of clothing (CLOT) and furniture (FURN). In addition, the paper also finds and acknowledges that the homogeneity restriction holds using the demand function for CLOT at 5% significance level. However, the homogeneity restriction condition does not hold based on the demand function for FURN at 5% significance level. The implication is that the negative effect of own price elasticity tends to even out the positive effect from the cross price elasticity for clothing products but not for furniture products. Finally, the paper concludes that the coefficients of the estimated OLS models for both CLOT and FURN seem to change with time due to structural change in the respective models.

References

Cameron, AC & Trivedi, PK 2014, *Microeconometrics using STATA*, London: MIT Press.

Chang, CL & McAleer, M 2015, ‘Econometric analysis of financial derivative’, *Journal of* *Econometrics*, vol. 187, no. 2, pp. 403-634.

Ghylsels, E & Marcellino, M 2016, ‘The econometric analysis of mixed frequency data sampling’, *Journal of Econometrics*, vol. 193, no.2, pp. 291-446.

Greene, WH 2017, *Econometric analysis*, New York: Pearson Education Publishers.

Gujarati, D & Porter, D 2016, *Basic econometrics*, New York: Irwin Publishers.

Pedace, R 2013, *Econometrics for dummies*, New York: Wiley & Sons Publishers.

Stock, JH & Watson, MW 2017, *Introduction to econometrics*, New York: Pearson Education Publishers.

Studenmund, AH 2017, *Using econometrics: a practical guide*, New York: Pearson Education Publishers.

Talukder, RK 2001, ‘Test of homogeneity condition in the demand for selected food items in Bangladesh’, *Bangladesh Journal of Agricultural Economics*, vol. 15, no.1, pp. 83-94.

Wooldridge, J 2016, *Introductory econometrics: a modern approach*, London: Cengage Publishers.