estimation of the household consumption

Econometrics for Economists

ECO00003I

Project: Estimation of the Household Consumption

Function

Exam No: Y6331815

Introduction and Description of the Economic Model:

In order to specify, estimate and interpret econometric models of the household consumption function using a sample from the cross-section Expenditure and Good Survey (2005-06) it would be prudent to explain the three main theories of consumption which would help us to specify our consumption model.

i) The Keynesian Consumption Function: The consumption function defined by Keynes describes the main component of GDP/Capita, which is consumption. Data and research act as evidence showing that the largest European countries account for about 60% of their total GDP spending on the component of consumption. (Blanchard) The main focus of the consumption function is determined by the disposable income (Yd) where disposable income is total income (Y) less tax (T). The consumption function is as follows:

C = c0 + c1 (YD)

Within the following layout, the marginal propensity to consume (c1) and the parameter (co) are taken into account as well. The marginal propensity to consume addresses effect a unit change on disposable income would bring on to the average consumption whereas the parameter of (c0) tends to recognise that individuals would consume (dissaving) some amount if their income is zero.i

ii) The Permanent Income Hypothesis: The Permanent Income Hypothesis is a theory on consumption, which was postulated by Friedman. The theory argues that the level of consumption of individuals depends on their “expected long-term average income.” This expectation of their long-term average income is therefore termed as the permanent income. Within this theory, it would be crucial to understand that an individual would resort to dissaving once their current level of income exceeds their expected long-term income vice-versa. Therefore, the framework of the theory pertains to the fact that a change in income would induce little or no change in the pattern of consumption, which is contrary to the theory postulated by Keynes.ii

iii) The Life-Cycle Hypothesis: The theory postulated by Modigliani, called the Life-Cycle Hypothesis pertains to the consumption and savings patterns of individuals across a lifetime. The theory argues that consumption, therefore, is fixed at a constant ratio of anticipated lifetime earnings of an individual. It should be noted that the life-cycle hypothesis leads to different behaviour of different age groups within a life cycle. An individual that is young, hence without sufficient skills to be employed to earn income, would have a higher propensity to consume than that of a middle-aged individual with a steady income. The individual

borrowing against future earnings can explain this. The same argument can be used for the retired individuals. However, these individuals’ propensity to consume would be higher as they would be dissaving over their past earnings. The middle-aged individuals would have a higher propensity to save and a lower one to consume, which would be propelled by a higher income than the other two categories.iii

A Description of the Model:

i) Ideal Consumption Specification:

By taking into consideration mentioned above, a model of consumption would be ideal if it were to contain the variables, which are the following:

a) Inflation:This explanatory variable, connected with price, would be used within the model to speculate whether the economic environment significantly affects the consumption behaviour of individuals. This variable would be negatively related to consumption.iv

b) Technology:

The efficient use of new technologies entering today’s market may account for the variation of consumption patterns within our model. E.G.: Efficient use of public transport enables lower usage of private mobilisation.

c) Unemployment Rate:The unemployment rate within the UK in that period would certainly play a huge role in dictating the patterns of consumption. The higher the unemployment rate, the lower the consumption.v

d) Wealth:Though not explicitly, the necessity of including the variable of wealth within the Modigliani consumption model is of paramount importance.vi

e) Interest Rate:The interest is not explicitly stated within the Keynesian Consumption function. However, its effect on demand seems to offer an explanation to the behaviour of consumption as well.vii

ii) Practical Consumption Specification: Include Marginal Propensity to Consume

Within the data given from the EFS 2005-06, the discussion of specification would revolve around the following variables:

a) The model does not contain the dummy variable of North Ireland as it is used as the reference variable and hence preventing the model from the dummy variable trap.

b) The model contains interaction effects on Linc with regions to show the differential slope between the region and base and between singleHH incomes and multiple income households.

c) Functional Form: The functional form taken within the present model is dictated by the fact that the elasticity of disposable income and weekly consumption is held constant within the semi-log model whereas within a purely linear model, the elasticity would change from point to point. This adheres to economic theory.

e) The choice of the model was taken by viewing the high t-values, adhering to the principle of Ockam Razor of parsimony, identifiability, predictive power and theoretical consistency.

A Discussion of any Data Issues, Limitations and Concerns:

There are essentially five main issues to consider when critiquing the model at hand.

i) Data Collection: Passive raw data collectors carry out data collection. These collectors are often ignorant to economic theory. In addition to this, most of the data collected takes place in an uncontrolled environment; therefore, it may include the effects of other factors as well.

ii) Out-Dated: The data collected within the current framework may be redundant in estimating and forecasting the future consumption patterns to the depreciation of time.

iii) Multicollinearity :Note that this is one of the assumptions of our CLRM and is a sample phenomenon. If our data shows exact or perfect multicollinearity, the hypothesis tests within our sample would lead to misleading conclusions, as the variance estimated would be higher than the true value.

iv) Heteroscedacticity: This is another assumption of the CLRM to be fulfilled. This problem is usually found in data of the cross-sectional category and hence we may be vary of this within estimating our models. This occurs when the members of our data may be of different sizes, such as different sizes of household. This leads to the variance of our error term to be non-constant and hence induces the model to the “scale problem.” Therefore, our estimators are no longer BLUE (Best Linear Unbiased Estimates) and our hypothesis tests become redundant as they observe misleading results due to the variance.

v) Autocorrelation: (spatial correlation) Another assumption of the CLRM where assumes that the members in our model are not correlated. e.g.: If one family were to increase consumption, another family might follow suit and hence we would have the problem of autocorrelation.E (uiuj) = 0 where ui ≠ uj (Assumption of CLRM Model)

Hypothesis Tests and Interpretation of Various Tests and Estimated Models:

Misspecification: We could detect the misspecification of the model in two main ways, by drawing a scatter-gram of the residuals on estimated Lexp in order to determine a systematic relationship and by conducting the RESET test by estimating a mode with the added powers of the fitted regression. After conducting the test, we found that the model was misspecified where our F-Value of 31.159 rejects critical F at 1% significance level, as the increase in R2 was statistically significant. Meaning our model may be prone to omission of a variable or the functional form may not be correct. The consequences of misspecification are the following:

1) Omission of Relevant Variable (Detail explained in the section of Omitted Variables below)

2) Incorrect functional form: Biased Estimators of true parameters.

Heteroscedasticity:

Consequences:

1) No longer BLUE (minimum variance does not hold)2) Biased variances of estimators3) Estimator of variance not holding4) Hypothesis Tests no longer hold

Chow-Test for Structural Stability:

Regional Differences in Household Consumption Levels:

Individual Significance:

Note slope coefficient = Linc in interaction analysisi) Linc (B2):

- Measures the elasticity of consumption expenditure with respect to disposable income

- For a percentage increase in inc, exp on average would increase by 0.375568% ceteris paribus.

- r2 = 0.0228- Significance: 1%- Magnitude/Sign: 0.375568/Positive

ii) workingHH:- Shows household with at least one member in work differs

from those with no workers in household positively by 0.181884 ceteris paribus.

- r2 = 0.0090- Significance: Reject (1%)- Economic Theory: Adhering to theory.

iii) North:

- Dummy variable showing Lexp in the North differs negatively from that in NI by -1.64559 ceteris paribus.

- r2 = 0.0108- Significance: Reject at 1%

iv) Midlands:- Dummy variable showing Lexp in Midlands differs negatively

from that in NI by -1.38200 ceteris paribus.- r2 = 0.0074- Significant: Reject at 1%

v) Scotland:- Dummy variable showing Lexp in Scot differs negatively from

that in NI by -1.50404- r2 = 0.0026- Significant at 1%.

vi) NI: (Base Constant)- Dummy variable of base category.- Shows the mean lexp in NI.- Magnitude: 3.24665 and positive- Significant: 1%

vii) Linc*South: (Differential Slope Coefficient)- Interaction effect showing by how much the slope coefficient

differs between the South and NI. - r2 = 0.0062- Significant: 1%- Sign/Magnitude: Positive/0.19937

viii) Linc*North: (Differential Slope Coefficient)- Interaction effect showing by how much the slope coefficient

differs between the North and NI. - r2 = 0.0099- Significant: 1% Level- Sign/Magnitude: Positive/0.263540

ix) Linc*Midlands:- Interaction showing by how much the slope coefficient differs

between the Midlands and NI.- r2 = 0.0067- Significant: 1% Level- Sign/Magnitude: Positive/0.218167

x) Linc*singleHH:- Differential slope coefficient showing by how much the slope

coefficient differs between single household income and multi household income in NI.

- r2 = 0.0109- Reject at 1%- Sign/Magnitude: Negative/-0.159865

xi) Variables that are not statistically significant:

- After testing each variable individually, we were able to assess that ageHH, ageHH2, singleHH, South, Wales, Linc*Wales and Linc*Scot were not statistically significant at the 1% level with the appropriate standard errors taken after testing for heteroscedasticity.

Errors of Measurement:

Causes:

1) Rounding Off2) Extrapolation3) Interpolation4) Smoothing of Data (Serving Hypothesis)

Consequences:

If measurement errors are present in our dependent variable, then the OLS estimators are unbiased along with their variances. However, the estimated variances are relatively larger as the independent variable gets added to the error term. Relatively, measurement errors in dependent variables do not matter much in practice.If measurement errors are present in explanatory variables, then the OLS estimators are biased and inconsistent in large samples.

Omitted Variable Bias:

If our model were to be under-fitted, it would produce biased estimates, as the excluded variables would be correlated with the included ones. Also, they would be inconsistent in large samples where the bias would not disappear. The error variance would also be biased along with the variance of estimated parameters by overestimating the true variance. All this would cause us to accept misleading hypothesis results given the data.

Conclusion:

After interpreting the model, the analysis can safely conclude that the coefficient associated with the Life-Cycle hypothesis (ageHH, ageHH^2) held little or no significance in our weekly consumption expenditure within the UK. Note however, that the data did not consist of observations over the age of 65, which might have had a significant impact in assessing the coefficient of ageHH.The function would’ve held more relevance within analysing consumption behaviour had it included variables of wealth as well. It provides a basic understanding of the structure of consumption in the UK and outside but without the relevant variable of wealth and more observations of retired individuals, it would result in misleading conclusion.

APPENDICES:

1) Alternative Models:

2) The Chow Test for Structural Stability:

H0: Model is structurally stableH1: H0 is false

S1 = RSSur (d.f. = N1 + N2 – k) = 0.507658 (d.f. = 1867 – (2)(4))

S2 = RSSr1 with (d.f. = N1 – k) = 0.372047 (d.f. = 1313 – 4)

S3 = RSSr2 with (d.f. = N2 – k) = 0.329385 (d.f. = 554 – 4)

S4 = RSSr1 + RSSr2 with d.f. = N1 + N2 – 2k

S5 = S1 – S4

Compute F = (S5 / k) ÷ [(S4) / (N1 + N2 – 2k)]

F = (-0.193774 / 4) ÷ (0.701432) / (1876 – 8)

F = -129.0110

Level of Significance ( )α 0.01 0.05 0.10Critical t-value 10.00 4.96 3.29

Reject H0 at 1% Level of Significance suggesting that the model is not structurally stable.

3) Testing Regional Differences: (Test of Overall Significance of a Multiple Regression Model in terms of R2)

H0: No significant regional difference between mean household consumption levelsH1: H0 is false

R r2 = 0.546096

F = (RUR2 – RR

2)/m ÷ [(1 – RUR2) / (n – k)]

F = [(0. 547851 – 0.546096)/6] ÷ [(1 – 0. 547851) / (1867 – 11)F = 1.2 where: F*m, n – k = F*6, 1867 – 11


Therefore: Cannot reject H0

This suggests that there does not appear to be some significant difference between regional mean household consumption levels.

4) Testing for Individual Significance: (Main Model)

Note: After conducting the heteroscedasticity test, we were able to find the robust standard errors corrected for heteroscedasticity to which the follow are tested with.

H0 = the dependant variables within the model are individually significantH1 = H0 is false

i) Testing Individual Significance of B1: Contant(Test of Significance Approach: Two-Tailed Test)

H0: B1 = 0H1: B1 ≠ 0

When σ2 is not known, then the test would follow the t-distribution.t = (b1 – B1) ÷ se (b1) ~ tn-k

t = (b1 – B1) ÷ se (b1) ~ t1867-11

t = (3.24665 – 0) ÷ 0.44756

|t| = 7.25

Reject H0 with 1% significance level.

ii) Testing Individual Significance of B2: Linc(Test of Significance Approach: Two-Tailed Test)

H0: B2 = 0H1: B2 ≠ 0


t = (b2 – B2) ÷ se (b2) ~ t1867-11

t = (0.375568 – 0) ÷ 0.063909|t| = 5.87

Reject H0 with 1% significance level.


iii) Testing Individual Significance of B3: singleHH(Test of Significance Approach: Two-Tailed Test)

H0: B3 = 0H1: B3 ≠ 0


t = (b3 – B3) ÷ se (b3) ~ t1867-11

t = (0.577137 – 0) ÷ 0.31451|t| = 1.84

Cannot reject H0 at 1% significance level.Reject H0 at 5% significance level.


iv) Testing Individual Significance of B4: ageHH(Test of Significance Approach: Two Tailed Test)

H0: B4 = 0H1: B4 ≠ 0


t = (b4 – B4) ÷ se (b4) ~ t1867-11

t = (0.010492 – 0) ÷ 0.010378

|t| = 1.01

Cannot reject H0 at 1% significance level.


v) Testing Individual Significance of B5: ageHH2(Test of Significance Approach: Two Tailed Test)

H0: B5 = 0H1: B5 ≠ 0


t = (b5 – B5) ÷ se (b5) ~ t1867-11

t = (-0.000122604 – 0) ÷ 0.00012327|t| = -0.9945

Cannot reject H0 at 1% significance level.


vi) Testing Individual Significance of B6: workingHH(Test of Significance Approach: Two Tailed Test)

H0: B6 = 0H1: B6 ≠ 0


t = (b6 – B6) ÷ se (b6) ~ t1867-11

t = (0.181884 – 0) ÷ 0.056202|t| = 3.23

Reject H0 at 1% significance level.


vii) Testing Individual Significance of B7: South(Test of Significance Approach: Two Tailed Test)

H0: B7 = 0H1: B7 ≠ 0

When σ2 is not known, then the test would follow the t-distribution.

t = (b7 – B7) ÷ se (b7) ~ tn-k

t = (b7 – B7) ÷ se (b7) ~ t1867-11

t = (-1.22097 – 0) ÷ 0.45505|t| = 2.68

Cannot reject H0 at 10% Level of Significance


viii) Testing Individual Significance of B8: North(Test of Significance Approach: Two Tailed Test)

H0: B8 = 0H1: B8 ≠ 0


t = (b8 – B8) ÷ se (b8) ~ t1867-11

t = (-1.64559– 0) ÷ 0.44338|t| = 3.71

Reject H0 at 1% Level of Significance


ix) Testing Individual Significance of B9: Midlands(Test of Significance Approach: Two Tailed Test)

H0: B9 = 0H1: B9 ≠ 0


t = (b9 – B9) ÷ se (b9) ~ t1867-11

t = (-1.38200– 0) ÷ 0.41449|t| = 3.33



x) Testing Individual Significance of B10: Wales(Test of Significance Approach: Two Tailed Test)

H0: B10 = 0H1: B10 ≠ 0


t = (b10 – B10) ÷ se (b10) ~ t1867-11

t = (-1.25767– 0) ÷ 0.65422|t| = -1.922

Cannot reject H0 at 1% Level of SignificanceReject H0 at 5% Level of Significance


xi) Testing Individual Significance of B11: Scot(Test of Significance Approach: Two Tailed Test)

H0: B11 = 0H1: B11 ≠ 0


t = (b11 – B11) ÷ se (b11) ~ t1867-11

t = (-1.50404– 0) ÷ 0.75025|t| = 2.00



xii) Testing Individual Significance of B12: Linc*South(Test of Significance Approach: Two Tailed Test)

H0: B12 = 0H1: B12 ≠ 0


t = (b12 – B12) ÷ se (b12) ~ t1867-11

t = (0.199377 – 0) ÷ 0.072649|t| = 2.74



xiii) Testing Individual Significance of B13: Linc*North(Test of Significance Approach: Two Tailed Test)

H0: B13 = 0H1: B13 ≠ 0


t = (b13 – B13) ÷ se (b13) ~ t1867-11

t = (0.263540 – 0) ÷ 0.071295|t| = 3.70



xiv) Testing Individual Significance of B14: Linc*Midlands(Test of Significance Approach: Two Tailed Test)

H0: B14 = 0H1: B14 ≠ 0


t = (b14 – B14) ÷ se (b14) ~ t1867-11

t = (0.218167 – 0) ÷ 0.066528|t| = 3.28



xv) Testing Individual Significance of B15: Linc*Wales(Test of Significance Approach: Two Tailed Test)

H0: B15 = 0H1: B15 ≠ 0


t = (b15 – B15) ÷ se (b15) ~ t1867-11

t = (0.191934 – 0) ÷ 0.10586|t| = 1.81



xvi) Testing Individual Significance of B16: Linc*Scot(Test of Significance Approach: Two Tailed Test)

H0: B16 = 0H1: B16 ≠ 0


t = (b16 – B16) ÷ se (b16) ~ t1867-11

t = (0.242004 – 0) ÷ 0.12135|t| = 1.99



xvii) Testing Individual Significance of B16: Linc*singleHH(Test of Significance Approach: Two Tailed Test)

H0: B16 = 0H1: B16 ≠ 0


t = (b16 – B16) ÷ se (b16) ~ t1867-11

t = (-0.159865 – 0) ÷ 0.053335|t| = 3.00



5) Testing for Elasticity:

H0: B2 = 1 (Unitary Elastic)H1: B2 ≠ 1 (H0 is false)

When σ2 is not known, then the test would follow the t-distribution.

t = (b2 – B2) ÷ se (b2) ~ tn-k

t = (b2 – B2) ÷ se (b2) ~ t1867-11

t = (0.375568 – 1) ÷ 0.01948|t| = 32.06

Reject H0 at 1% Level of Significance.


Elasticity Elasticity Coefficient

Inelastic Absolute value less than 1Elastic Absolute value greater than 1Unitary Value equal to 1

6) Testing for Misspecification (RESET Test):

H0: Model is correctly specified

H1: Model is incorrectly specified

F = [(R2new – R2

old) / Number of New Regressors] ÷ [(1 – R2new) / (n – number of

parameters in the new model)

F = (0.571774 – 0.557333) / 2 ÷ (1 – 0. 571774) / 1867 – 19)

F = 31.159

The increase in R2 is statistically significant.The computed F-Value we obtained is statistically significant at 1% level.


Therefore, we reject H0 that the model is correctly specified.

7) Testing for Heteroscedasticity (White’s Test):

H0: No heteroscedasticity exists

H1: H0 is false

n*R2 ~ ²χ k-1

(1867)(0.132356)

²χ 24 = 247.11

Hence, we reject H0 suggesting that heteroscedasticity may exist at 1% Level of Significance.

Level of Significance ( )α 0.01 0.05 0.10Critical ²χ 24 42.9798 36.4151 33.1963

Omission of Relevant Variables (Under-fitting a Model):

1) Biased Estimates:Omission of a relevant variable leads to the correlation of the excluded variable with the included one.Results in the estimated values of the parameters not reflecting the true value on average.E.G:Yi = B1 + B2X2i + B3X3i + ui (Correctly Specified model)Yi = A1 + A2X2i + vt (Under-fitted Model)Therefore:E (a1) ≠B1 and E (a2) ≠B2

E (a1) = B2 + B3b32 and E (a2) = B1 + B3 (x̄3 – b32 x̄2)Where b32 is the slope coefficient of the omitted variable X3 on the included X2. Therefore X2 represents the direct effect on Y and also the indirect effect of X3 on Y.If both B3 and b32 are positive then it would represent an upward bias which means that it overstates the true parameter value.

2) Inconsistent:a1 and a2 inconsistent.Meaning no matter how large a sample gets, the bias would not disappear.

3) Error Variance Biased:Where E(var) ≠True Variance

4) Variance of Estimator Biased:Biased even when the two explanatory variables are uncorrelated.Var (a2) ≠Var (b2)

5) Confidence Interval and Hypothesis Testing:They become unreliable as the confidence interval become much larger.

i Friedman, M. (1957), “A Theory of the Consumption Function” http://www.nber.org/chapters/c4403.pdf (Accessed 2 April 2012)ii Friedman, M (1957) “The Permanent Income Hypothesis” http://www.nber.org/chapters/c4405.pdf (Accessed 2 April 2012)iii Deaton, A. (2005), Franco Modigliani and the Life Cycle Theory of Consumption, http://www.princeton.edu/~deaton/downloads/romelecture.pdf (Accessed 2 April 2012)ivPoole, W. (2001), The Role of Interest Rates and Inflation in Consumption Function, http://www.brookings.edu/~/media/Files/Programs/ES/BPEA/1972_1_bpea_papers/1972a_bpea_poole.pdf (Accessed 2 April 2012)v Malley, J. (1996), Unemployment and Consumption, http://www.jstor.org/stable/2663613 (Accessed 2 April 2012)vi Evans, M. (1995), The Importance of Wealth in the Consumption Function, http://www.jstor.org/stable/1828595 (Accessed 2 April 2012)vii Gujarati, D. and Porter, C.D. (2010). Essentials of Economics, McGraw Hilli Kapoor, M. (2008), Effect of Interest Rate on Consumption, (http://www.isid.ac.in/~pu/conference/dec_08_conf/Papers/ShamikaRavi.pdf (Accessed 2 April 2012)ix Gujarati, D. and Porter, C.D. (2009). Basic Econometrics, McGraw Hillx J. Hey, Intermediate Microeconomics, (1st Edition), Pearson, 2010xi O. Blanchard, A Amighini & F. Giavazzi, Macroeconomics: A European Perspective, (1st Edition), Pearson, 2010

http://www.isid.ac.in/~pu/conference/dec_08_conf/Papers/ShamikaRavi.pdf

http://www.jstor.org/stable/1828595

http://www.jstor.org/stable/2663613

http://www.brookings.edu/~/media/Files/Programs/ES/BPEA/1972_1_bpea_papers/1972a_bpea_poole.pdf

http://www.brookings.edu/~/media/Files/Programs/ES/BPEA/1972_1_bpea_papers/1972a_bpea_poole.pdf

http://www.princeton.edu/~deaton/downloads/romelecture.pdf

http://www.nber.org/chapters/c4405.pdf

http://www.nber.org/chapters/c4403.pdf

estimation of the household consumption

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