Quadratic Food Engel Curves with Measurement Error: Evidence from a Budget Survey

Transcription

1 See discussions, stats, and author profiles for this publication at: Quadratic Food Engel Curves with Measurement Error: Evidence from a Budget Survey ARTICLE in OXFORD BULLETIN OF ECONOMICS & STATISTICS FEBRUARY 2007 Impact Factor: 1.37 DOI: /j x Source: RePEc CITATIONS 12 READS 65 2 AUTHORS: Sourafel Girma University of Nottingham 136 PUBLICATIONS 3,783 CITATIONS Abbi Kedir University of Leicester 7 PUBLICATIONS 53 CITATIONS SEE PROFILE SEE PROFILE Available from: Sourafel Girma Retrieved on: 09 April 2016

2 OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 69, 1 (2007) doi: /j x Quadratic Engel Curves with Measurement Error: Evidence from a Budget Survey* Abbi Kedir and Sourafel Girma Department of Economics, University of Leicester, Leicester, UK ( ak138@le.ac.uk) Nottingham University Business School, Industrial Economics Division, Nottingham, UK ( sourafel.girma@nottingham.ac.uk) Abstract This paper examines the importance of accounting for measurement error in total expenditure in the estimation of Engel curves, based on the 1994 Ethiopian Urban Household Survey. Using Lewbel s [Review of Economics and Statistics (1996), Vol. 78, pp ] estimator for demand models with correlated measurement errors in the dependent and independent variables, we find robust evidence of a quadratic relationship between food share and total expenditure in the capital city, and significant biases in various estimators that do not correct for correlated measurement errors. I. Introduction Engel curve analysis has been an important tool in understanding the dynamics of household welfare. For example it has proved useful in the modelling of income distribution and the evaluation of indirect tax policy reform (Banks, Blundell and Lewbel, 1997; Gibson, 2002; You, 2003). Many studies employed the Working- Leser specification in which budget shares are assumed to be linear functions of total expenditure (Leser, 1963; Deaton and Muellbauer, 1980). However, there is now a *We would like to thank the editor (Jonathan Temple), and the anonymous referees for constructive comments, which have substantially improved the paper. We have also benefited from discussions with Marcel Fafchamps, Sonia Bhalotra, other participants at the conference of the Centre for the Study of African Economies, University of Oxford in April 2004, Derek Deadman and Steve Wheatley Price. Thanks are also due to the Department of Economics of Addis Ababa University for allowing us access to the data collected by the Ethiopian Urban Household Survey. JEL Classification numbers: C14, C21, D12, I Blackwell Publishing Ltd, Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

3 124 Bulletin growing body of empirical evidence that documents robust nonlinear relationships in budget share equations, particularly for non-food items (Lewbel, 1991; Hausman, Newey and Powell, 1995; Banks et al., 1997). 1 In this study, we show that nonlinear Engel curves can also be observed for food items within the context of a developing country. To be specific, we estimate quadratic Engel curves for food items using the 1994 Ethiopian Urban Household Budget Survey. As Lewbel (1996) convincingly argues, when total expenditure is measured with error, the share of expenditure on an individual good (which is the dependent variable in demand models) is also likely to be measured with error, and the measurement errors in the dependent and independent variables need not be independent. As a result, demand estimation with errors in total expenditure cannot be handled using standard instrumental variables techniques. Thus we apply Lewbel s (1996) two-step approach which allows for correlated measurement errors in the food budget share and total household expenditure. The first step of this approach multiplies the original demand equation by different powers of the observed total expenditure and then uses a generalized method of moments (GMM) technique to obtain consistent estimates of the transformed model. The second step exploits the functional relationships between the parameters of the demand model and the parameters of the transformed model to recover consistent estimates of the former. To the best of our knowledge, this is the first paper to do so using developing country data where measurement error problems are arguably more severe (Deaton, 1997; Gibson, 2002). To emphasize the importance of correcting for measurement errors in budget shares, we also apply the instrumental variables procedure suggested by Hausman et al. (1995) that neglects measurement error in the left-hand-side variable. The paper is organized as follows. Section II reviews the literature on Engel curve analysis with measurement errors. Section III introduces the quadratic errorsin-variables inference procedure. Section IV discusses the data used in our empirical analysis and reports results from a preliminary data analysis. The main findings of the paper are discussed in section V. Finally, section VI concludes. II. Literature The presence of measurement error has been duly recognized in the literature on Engel curve estimation since the early 1960s. However, the treatment of measurement error in quadratic Engel curves is quite recent and is often undertaken using data sets from Organization for Economic Co-operation and Development (OECD) countries and on non-food items for which the quadratic form is well established. Based on evidence from a household expenditure survey data for Israel, Liviatan (1961) established that neglected measurement error induces non-negligible bias in ordinary least 1 See also Delgado and Miles (1997), Gozalo (1997) and You (2003) for applications of parametric and nonparametric methods in the estimation of nonlinear Engel curves.

4 Quadratic Engel curves with measurement error 125 square (OLS) estimates of linear Engel curve parameters. Econometric advances in the early 1990s, by Hausman et al. (1991), developed a root-n consistent estimator for polynomial specifications with measurement error only in the right-handside expenditure variable of the demand equation.aasness, Biørn and Skjerpen (1993) explicitly modelled measurement error in the estimation of a system of consumer functions from Norwegian household budget data, and concluded that measurement error accounted for 27% of the variability of the observed total consumption expenditure. Hausman et al. (1995) employed a nonlinear error-in-variables model to examine the parameters of some Engel curves using US Consumer Expenditure Survey (CES) data, and estimated that 42% of the total variation in measured expenditure was due to measurement error. Using data from the British Family Expenditure Survey, Hasegawa and Kozumi (2001) considered Engel curve estimation with measurement error from a Bayesian perspective, and reported that the observed mean household total expenditure overestimated the mean of the true total expenditure. Under the assumption of classical measurement error with a normally distributed regressor, Kuha and Temple (2003) have shown that OLS-type naïve estimators tend to generate estimates where the quadratic function appears to curve less steeply than it actually does. Further evidence from the UK is provided by Banks et al. (1997), who found the presence of a quadratic relationship between expenditure on nonfood items and total household expenditure. However, the study could not reject the Working-Leser linear specification for food. In a recent application on budget survey data from Canada, You (2003) showed that robust estimators point to lower income elasticities and have better performance than the standard least squares and Tobit estimators when there are obvious outliers in the data. Betti (2000) showed evidence of nonlinearities in Engel curves for seven commodities including food using Italian budget data, but did not control for measurement error. Bhalotra and Attfield (1998) estimated semiparametric Engel curves for rural Pakistan using a large household survey. The authors main objective was to investigate whether the distribution of resources within families tends to favour males over females, older over younger children, or prime-age over elderly adults. They were also motivated by the need to establish the shape of the Engel curve. Their findings indicate that there is no differential treatment of the elderly or higher birth-order children in resource allocation. They identified substantial economies of size in food consumption. Interestingly, they also find that Engel curves for food, adult goods and child goods are quadratic logarithmic, which implies that some commonly used demand models (e.g. piglog) are inappropriate. Within a framework of household demand equations for fuel in the United Kingdom, Lewbel (1996) found that correction for measurement error changed the parameter estimates by more than 15%. This study is unusual because it controls for measurement errors both on the left- and right-hand side of the demand equation. In general, there is a dearth of studies, especially for developing countries, that analyse Engel curve relationships in conjunction with the treatment of measurement error in the expenditure variable.

5 126 Bulletin III. Estimating quadratic Engel curves with measurement errors Let x Å h and y Å h be the true total consumption expenditure and true expenditure on food items, respectively, for household h = 1,..., H. Moreover, let x h and y h be the measured counterparts of x Å h and y Å h, respectively, and let the true and measured food share in consumption be w Å = yå h h x Å and w h = y h, h x h respectively. Define the measurement error in x h as V h = x h /x Å h so that ln V h = ln x h ln x Å h. (1) Similarly, define the measurement error in the observed food budget share as Now consider the following quadratic food share model ω h = w h w Å h. (2) w Å h = a + b ln xå h + c(ln x Å h ) 2 + u h, (3) where u h is a mean zero error. Combining the behavioural model (3) with the measurement models (1) and (2) gives the estimating equation w h = a + b ln x h + c(ln x h ) 2 + ε h, (4) where, by definition, the composite error term ε h can be expressed as ε h = ω h + u h b ln V h + c(ln V h ) 2 2c(ln V h ln x h ). (5) Even if the standard assumption that the model error u h is independent of the measurement errors ln V h and ω h holds, we need to control for measurement errors both in the left- and right-hand sides of equation (4). This is necessitated by the fact that the observed food consumption y h is part of observed total consumption x h (Lewbel, 1996). Hence, by definition, measurement error in the latter is related to measurement error in the former. Lewbel (1996) proposes a two-step approach that delivers consistent estimators of the parameters of the demand models. The first step consists of multiplying the original demand equation by different powers of the observed total expenditure (in levels and not in logs), and then applying instrumental variables or a GMM estimation technique to obtain consistent estimates of the transformed model. The second step exploits the relationship between the original parameters and the parameters of the transformed model to recover consistent estimates of the former. In what follows we describe briefly how we have implemented this estimation strategy. We first multiply equation (4) by x h and follow Lewbel (1996, section VI) in using household income, income squared, and the interaction terms between income and log of income as instrumental variables, denoted by z h, in the transformed model.

6 Quadratic Engel curves with measurement error 127 Under the assumption that the instruments are valid, E(z h x h ε h ) = 0, equation (32) in Lewbel (1996) with k = 1, shows that the following moment conditions hold: E z h (x h w h αx h βx h ln x h γx h (ln x h ) 2 ) = 0. (6) The parameters of the transformed model α, β and γ are estimated via GMM and the parameters of the Engel curve model (4) are recovered and their standard errors calculated by making use of the following relationships (Lewbel, 1996, Corollary 1, p. 723): a = α + βe(v h ln V h ) + γe[v h (ln V h ) 2 ] (7) b = β + 2γE[V h ln V h ], (8) c = γ. (9) It is clear from these relationships that the Lewbel approach depends on the knowledge of certain moments of the distribution of the measurement error ln V h. For this, one needs to assume either a specific distribution for the measurement error or use a nonparametric approach to estimate E(V h ln V h ) and E[V h (ln V h ) 2 ]. We estimate these moments nonparametrically using a series-based approximation described in equations (28) and (29) of Lewbel (1996). IV. The data Data description and preliminary analysis Our empirical analysis is based on the 1994 socioeconomic survey of urban households in Ethiopia. The survey includes questions on household demographics including education, rural urban migration, employment and income, consumption, ownership of durables, housing, health, welfare and welfare change indicators. A sample of 1,500 households was selected from seven major urban centres of the country. These are Mekele and Dessie in the north, Bahir Dar in the north-west, Addis Ababa in the centre, Dire Dawa in the east, Awassa in the south and Jimma in the south-west. Mekele and Dessie were selected to represent the socioeconomic groups in the north and areas often affected by drought. Bahir Dar was included as a representative town in the main cereal-producing areas of the country. Addis Ababa is by far the largest city and the capital, and represents the diversity of the country s population, because different groups of people from different parts of the country migrate to the capital in search of a better life. Dire Dawa is mainly a trading centre, while Awassa is the administrative centre of the south, and was chosen to represent the large Enset culture. 2 Finally, Jimma was selected to represent the urban characteristics of the main coffee-growing regions of the country. The total sample size was distributed over the selected urban centres proportionally to their populations, based on the Central Statistical Authority s population figure projections. Accordingly, the 2 This is one of the major food cultures in southern Ethiopia. Enset is often referred to as false banana.

7 128 Bulletin TABLE 1 Descriptive statistics for the food share data Addis Ababa Other urban areas Variable Mean Std. dev. Mean Std. dev. Food share Real total , expenditure Real income ,100.5 Household size Sample size Notes: The income and expenditure figures are reported in Ethiopian birr (ETB). The current official exchange rate is 1 = ETB as of 14 June sample included 900 households from Addis Ababa, 125 from Dire Dawa, 75 from Awassa and 100 from each of the other four towns. To show the regional variation in the food share expenditure relationship, we classified our total sample into households from the capital city (Addis Ababa) and households from other cities (Other Urban Areas). Traditional Engel curve analysis assumes that all households in a given survey face the same prices. We relax the usual constant price assumption in the present paper and apply carefully constructed spatial price deflators to the raw expenditure data. 3 We believe that the results for Addis Ababa are more reliable than those for the Other Urban Areas because the latter aggregates six different cities that may differ in their prices. Table 1 provides some summary statistics of the variables used in the study for households where complete information is available. 4 On average, the food share for Other Urban Areas is higher by about 3.5 percentage points compared with Addis Ababa, and this proves to be significant at the 1% level using a t-test for the equality of mean values. By contrast, no appreciable difference in the mean values of deflated total expenditure is found at conventional significance levels. Preliminary analysis using nonparametric and semiparametric models By way of preliminary analysis, we estimated some nonparametric and semiparametric Engel curves. Nonparametric and semiparametric regression analyses provide attractive alternatives to linear regression because they allow the data to determine the local shape of the conditional mean relationship. 3 Details on how they are constructed under different price assumptions can be found in Kedir, Disney and McKay (2003). We use unit values (i.e. the ratio of total expenditure to quantity) which are defined within the survey as prices. They are contaminated by measurement error and quality effects as argued in Deaton (1997). The unit values we have used for constructing the indices above are corrected in the sense that they are treated both for quality effects and measurement error. 4 Apart from missing observations, we also dropped households with food share values of 0% or 100%.

8 Quadratic Engel curves with measurement error Addis Ababa 1 Other urban areas Food share Food share Log expenditure Log expenditure Actual values Lower bound Upper bound Smooth fit Actual values Lower bound Upper bound Smooth fit Figure 1. Nonparametric kernel fits with 95% pointwise confidence interval: whole sample 1 Addis Ababa 1 Other urban areas Food share Food share Log expenditure Log expenditure Actual values Lower bound Upper bound Smooth fit Actual values Lower bound Upper bound Smooth fit Figure 2. Nonparametric kernel fits with 95% pointwise confidence interval: restricted sample We started by running nonparametric regressions of food share on the log of total expenditure. The smoothed food share values (bandwidth = 0.8) 5 along with 95% pointwise confidence intervals are sketched against actual expenditures in Figures 1 and 2. These figures show clear evidence of a quadratic relationship between food 5 We also experimented with various bandwidths, and the shape of the relationship remains unchanged.

9 130 Bulletin 0.8 Addis Ababa 0.75 Other urban areas Food share 0.4 Food share Log expenditure Log expenditure Endogenous Exogenous Endogenous Exogenous Figure 3. Semiparametric models of food share: whole sample share and real total expenditure, and this quadratic relationship does not appear to be driven by outliers. It is also apparent that the quadratic relationship is more pronounced in the case of other urban areas relative to the sample drawn from the capital city, Addis Ababa. We also adopt the partially linear model of Robinson (1988), in which we leave the functional relationship between food share and total expenditure unspecified, while a linear relationship between household size and food share is assumed. Thus the semiparametric regression specification has the following form: y i = g(z i ) + r i + ε i, (10) where the form of the function g( ) is unknown. The measurement error in the expenditure variable leads to a correlation between the disturbance of the food share equation and the expenditure variable. This gives rise to a potential endogeneity problem. For this reason, we also correct for this potential endogeneity problem following the methodology suggested by Blundell and Duncan (1998, pp ). 6 This requires the existence of suitable instruments for total expenditure, and we use total income and a dummy for the household head s gender as predictors of expenditure. 6 Bhalotra and Attfield (1998) followed a similar procedure for their study of Pakistan. In our case, we regressed (parametrically) expenditure on income (the instrument) and the head s gender dummy. The residual from this regression is included as an additional variable in the semiparametric model. Note that this procedure corrects for general endogeneity problems, and not just measurement errors.

10 Quadratic Engel curves with measurement error 131 Addis Ababa Other urban areas Food share 0.2 Food share Log expenditure Log expenditure Endogenous Exogenous Endogenous Exogenous Figure 4. Semiparametric models of food share: restricted sample The semiparametric regression results are presented in Figures 3 and 4, and they are quite consistent with the nonparametric fits discussed earlier. Distinct nonlinear behaviour in food share is found, especially for the data from Other Urban Areas. In addition, the assumption of exogenous real total household expenditure does not appear to have any substantial effect on the shape of the food Engel curve. To summarize, the nonparametric and semiparametric regressions reveal an approximate inverted U-shaped relation between budget shares for food and log of real total expenditure. This makes it apparent that the Leser-Working linear formulation is not an accurate approximation for our data. V. Quadratic Engel curve estimates The estimation results of Engel curves are reported in Tables 2 to 6. Overall, the findings suggest that the incorporation of the square of the log of household expenditure in our Engel curve equation, and the explicit consideration of measurement errors both in the budget share and total expenditure variables are important to arrive at reliable parameter estimates of the demand equation. This is particularly evident from the comparison of our results reported in Tables 4 and 6. First, we discuss the results from some preliminary regressions, to provide some benchmark results. We then discuss our findings from the measurement error-corrected model.

11 132 Bulletin TABLE 2 OLS, outlier robust and median regression estimates of the quadratic food Engel curve parameters: whole sample Addis Ababa Other urban areas OLS with robust SE Outlier robust Median regression OLS with robust SE Outlier robust Median regression Log of real total expenditure (4.48)** (6.43)** (3.96)** (3.09)** (3.52)** (4.72)** Square of log of real total (5.39)** (7.61)** (4.95)** (3.34)** (3.75)** (4.97)** expenditure Log of household size (2.97)** (2.92)** (3.67)** (0.22) (0.54) (0.00) Constant (0.55) (1.47) (0.10) (0.25) (0.37) (1.02) Turning point (95% confidence ( ) ( ) ( ) ( ) ( ) ( ) interval) Observations Notes: The dependent variable is the share of the household budget devoted to food. Absolute values of t-statistics are presented in parentheses. Significance at the 1% and 5% levels is indicated by ** and * respectively. The (asymptotic) confidence intervals for turning points are estimated using the delta method.

12 Quadratic Engel curves with measurement error 133 TABLE 3 OLS, outlier robust and median regression estimates: restricted sample Addis Ababa Other urban areas OLS with robust SE Outlier robust Median regression OLS with robust SE Outlier robust Median regression Log of real total expenditure (6.61)** (7.80)** (5.23)** (6.16)** (6.47)** (7.46)** Square of log of real total expenditure (7.72)** (9.26)** (6.45)** (6.29)** (6.70)** (7.68)** Log of household size (3.93)** (3.97)** (4.05)** (0.32) (0.28) (0.22) Constant (2.03)* (2.31)* (0.87) (2.63)** (2.88)** (4.23)** Turning point [95% confidence interval] [ ] [ ] [ ] [ ] [ ] [ ] Observations Notes: The dependent variable is the share of the household budget devoted to food. Absolute values of t-statistics are provided in parentheses. Significance at the 1% and 5% levels is indicated by ** and * respectively. The (asymptotic) confidence intervals for turning points are estimated using the delta method. The top and bottom 5% observations in terms of standardised residuals are omitted in the restricted sample.

13 134 Bulletin TABLE 4 Estimates of the quadratic Engel curve parameters based on Lewbel (1996) Addis Ababa Other urban areas Whole sample Restricted sample Whole sample Restricted sample Log of real total expenditure (3.84)*** (15.68)*** (3.11)*** (4.11)*** Square of log of real total expenditure (4.34)*** (20.19)*** (2.92)*** (4.86)*** Log of household size (2.59)*** (5.29)*** (4.46)*** (1.32) Constant (2.61)*** (8.88)*** (3.56)*** (1.21) Turning point [95% None 332 confidence interval] [ ] [ ] [ ] Number of observations Notes: The dependent variable is the share of the household budget devoted to food. Absolute values of t-statistics are provided in parentheses. Significance at the 1% and 5% levels is indicated by ** and * respectively. The (asymptotic) confidence intervals for turning points are estimated using the delta method. The top and bottom 5% observations in terms of standardized residuals are omitted in the restricted sample. TABLE 5 Some expenditure elasticity estimates Addis Ababa Other urban areas Expenditure percentile OLS Lewbel (1996) OLS Lewbel (1996) 5th (0.016) (0.051) (0.019) (0.023) 10th (0.012 (0.039) (0.015) (0.019) 25th (0.089) (0.011) (0.009) (0.004) 50th (0.007) (0.001) (0.009) (0.007) 75th (0.074) (0.033) (0.012) (0.028) 95th (0.013) (0.057) (0.019) (0.044) Notes: Standard errors are given in parentheses. Calculations are based on the restricted sample. Throughout we generate estimates based on the whole sample and on a restricted sample obtained by removing outliers. 7 An observation is identified as an outlier if 7 We thank the reviewers for pointing out that the quadratic relationship in our data might have been driven by outlying observations. During our initial exploratory analysis, we also investigated if there is a need for a further (i.e. third order) polynomial term, but this notion is decidedly rejected by the Ramsey RESET test and other approaches. The quadratic relationship is found to be a dominant and appropriate specification for the food share equation we have investigated.

14 Quadratic Engel curves with measurement error 135 TABLE 6 Measurement error corrected IV estimates of the quadratic food Engel curve parameters (Hausman et al., 1995) Addis Ababa Other urban areas Restricted Restricted Variable Whole sample sample Whole sample sample Log of real total expenditure (2.36)* (6.79)** (4.59)** (8.74)** Square of log of real total expenditure (2.56)* (7.45)** (4.35)** (8.52)** Log of household size (1.64) (2.81)** (2.70)** (8.52)** Constant (0.48) (3.24)** (2.11)* (1.51) Turning point [95% confidence interval] [ ] [ ] [ ] [ ] Number of observations Notes: The dependent variable is the share of the household budget devoted to food. Absolute values of t-statistics are provided in parentheses. Significance at the 1% and 5% levels is indicated by ** and * respectively. The (asymptotic) confidence intervals for turning points are estimated using the delta method. The top and bottom 5% observations in terms of standardized residuals are omitted in the restricted sample. it lies below the 5th percentile or above the 95th percentile of the Studentized residual distribution. 8 Before presenting the measurement error-corrected results, we discuss the main findings from some preliminary regressions based on three alternative methods. These are the OLS (conditional mean) model with robust standard errors, outlier robust regressions (cf. Rousseeuw and Leroy, 1987), and median regressions. Tables 2 and 3 report the results from these preliminary regressions for the whole and restricted sample, respectively. We find that, irrespective of method and sample, the quadratic relationship is robust and statistically significant. However, the estimated turning points and their associated confidence intervals are sensitive to the estimation techniques adopted. For example, the median regression estimates imply much higher turning points for Other Urban Areas. The measurement error corrected estimates based on Lewbel s approach are presented in Table 4, and, except for the case of Other Urban Areas in the presence of outliers, they are supportive of the presence of quadratic Engel curves. In what follows, we confine our discussion to results from the restricted sample in which gross outliers are omitted. Starting our discussion with the sample from Addis Ababa, the turning point of the Engel curve is at 351 Birr. This value corresponds to the 63rd percentile of the expenditure data, and suggests that about two-thirds of the observations in the food 8 This is only a working definition that we have adopted. As argued by Temple (2000), among others, there is always an element of arbitrariness in attempts to identify outliers.

15 136 Bulletin Engel curve are sloping upward. It is noteworthy that the turning points implied by the various estimators that do not correct for measurement errors are much lower than 351 Birr. This is consistent with the notion that measurement error flattens the curvature of OLS-estimated quadratic functions and demonstrates the importance of correcting for measurement error. 9 Notice that the confidence intervals of the turning points associated with the measurement error corrected estimators are wider. This is a manifestation of the well known bias versus variance trade off problem, where the very process of correcting for bias makes the corrected estimator more variable than the biased estimator (cf. Carroll, Ruppert and Stefanski, 1995, section 2.4). Incidentally, compared to OLS, the measurement error corrected IV estimator yields turning points closer to the nonparametric regressions. We also find evidence of quadratic Engel curves among households outside the capital, although this is confined to the restricted sample where gross outliers are removed. The point estimate of the turning point of the Engel curve is 332 Birr, with a 95% confidence interval running from 176 to 490 Birr, which is wide.as highlighted earlier, we believe that the result for Addis Ababa is more reliable than for the sample outside Addis Ababa, since the latter aggregates several regions in which consumers are likely to face different relative prices. Table 5 gives expenditure elasticity estimates at some points of the expenditure distribution, and illustrates the importance of correcting for measurement errors in analysing budget shares. An alternative approach: the Hausman et al. (1995) estimator Hausman et al. (1995) proposes a two-step approach that delivers consistent estimators of the parameters of polynomial Engel curves with measurement errors. This procedure addresses the measurement error problem in the right-hand-side variable (i.e. total expenditure) only. To implement the approach, first one should multiply the demand equations by different powers of the observed total expenditure and then apply minimum variance instrumental variable techniques to obtain consistent estimates of the transformed model. The approach then exploits the relationship between the original parameters and the parameters of the transformed model to recover consistent estimates of the former via minimum chi-square problem techniques. 10 In this paper we implement the Hausman et al. (1995) approach by using the log and log square of income, the gender of the head of household and regional dummies as instruments. The results from these experiments are reported in Table 6. They show the presence of an inverted U-shaped relationship between food share and the 9 In a less complicated set up than considered here, Kuha and Temple (2003) have shown that one of the effects of measurement error is to flatten the curvature of the estimated functions. Note that our results do not necessarily follow those derived in their work mainly because we have controlled for measurement errors both in the budget share and the right hand side expenditure variables. In addition to the flattening point, Kuha and Temple (2003) discussed the impact that measurement error has on the observed turning point in a quadratic framework. In the simple case, they illustrated that the effect on the turning point depends on the location of the true turning point relative to the population mean of the true predictor (i.e. true log expenditure in our application). 10 For the sake of brevity, we have omitted details of the estimation procedure. See Kedir and Girma (2003) for details.

16 Quadratic Engel curves with measurement error 137 log of total expenditure in all subsamples. Comparison of the results in Tables 4 and 6 reveals some important differences between the two sets of results. For instance, the turning points for the Addis Ababa sub-sample are lower when one uses the Hausman et al. (1995) estimator. It is worth pointing out that the Hausman et al. (1995) estimator has two potential shortcomings. First, it need not be efficient as it is based on a limited set of moment restrictions. 11 Secondly, and potentially more seriously, it ignores the possibility that the food budget share is also likely to be measured with error as long as total expenditure is error-ridden. 12 VI. Conclusions This paper examined the importance of accounting for measurement errors in total expenditure in the estimation of Engel curves using data from the 1994 Ethiopian Urban Household Survey. To this end, it adopted Lewbel s (1996) estimator for a demand model with correlated measurement errors both in the dependent and independent variables. In the sample of households from the capital city (i.e. Addis Ababa) which constitute 61% of the total number of households in our study, we find a robust and statistically significant quadratic relationship between food share and total expenditure. At lower levels of expenditure, the Engel curve is upward sloping and we calculate that it starts to slope downward at the 63rd percentile of the total expenditure distribution. Interestingly, the Hausman et al. (1995) estimator also suggests the existence of quadratic Engel curves, but with much lower turning points than those implied by the Lewbel (1996) estimator. This demonstrates the importance of accounting for measurement errors in both budget share and total expenditure variables. By contrast, the evidence in favour of quadratic Engel curves for the samples from the various cities outside the capital is not as robust, and the measurement errorcorrected turning point is less precisely estimated. It is worth investigating in the future whether this is due to the fact that the samples outside Addis Ababa aggregate various regions in which consumers face different relative prices. Final Manuscript Received: November 2006 References Aasness, J., Biørn, E. and Skjerpen, T. (1993). Engel functions, panel data and latent variables, Econometrica, Vol. 61, pp Banks, J., Blundell, R. and Lewbel, A. (1997). Quadratic Engel curves and consumer demand, The Review of Economics and Statistics, Vol. 79, pp We thank a referee for pointing this out to us. 12 Note that our data are based on reported expenditure levels. However, one can envisage a situation in which households might not be accurate enough about their total expenditure or food expenditure, but reasonably accurate about the percentage of expenditure devoted to food.

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