Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: partial adjustment model Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 11). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

PARTIAL ADJUSTMENT 1 The idea behind the partial adjustment model is that, while a dependent variable Y may be related to an explanatory variable X, there is inertia in the system and the actual value of Y t is a compromise between its value in the previous time period, Y t–1, and the value justified by the current value of the explanatory variable.

PARTIAL ADJUSTMENT 2 Let us denote the justified value of Y (or target, desired, or appropriate value, however you want to describe it) as Y t *, given by the equation shown.

PARTIAL ADJUSTMENT 3 In the partial adjustment model it is assumed that the actual increase in the dependent variable from time t – 1 to time t, Y t – Y t–1, is proportional to the discrepancy between the justified value and the previous value, Y t * – Y t–1.

PARTIAL ADJUSTMENT 4 is usually described as the speed of adjustment.

PARTIAL ADJUSTMENT 5 The actual value in the current time period is therefore a weighted average of the desired value and the previous actual value. logically should lie in the interval 0 (no change at all) to 1 (full adjustment in the current time period).

PARTIAL ADJUSTMENT 6 Substituting for Y t * from the original relationship, one obtains a regression specification in terms of observable variables of the ADL(1,0) form. where

PARTIAL ADJUSTMENT 7 It follows that its dynamics are those of the ADK(1,0) model discussed in the previous slideshow. The short-run impact of X on Y is given by the coefficient  2 =  2.

PARTIAL ADJUSTMENT 8 The long-run effect can be evaluated by finding the relationship between the equilibrium values of Y and X.

PARTIAL ADJUSTMENT 9 The long-run effect turns out to be  2. This makes sense, since this is the coefficient in the equation determining the desired value of Y.

10 PARTIAL ADJUSTMENT Brown's Habit Persistence Model of the aggregate consumption function was an early example of the use of a partial adjustment model. Desired consumption is related to wage income, nonwage income and a dummy variable.

11 PARTIAL ADJUSTMENT The reason for separating income into wage income and nonwage income is that the marginal propensity to consume is likely to be higher for wage income than for nonwage income.

12 PARTIAL ADJUSTMENT Brown fitted the model with a time series which included observations before and after the Second World War. The dummy variable, A, was defined to be 0 for the prewar observations and 1 for the postwar ones.

PARTIAL ADJUSTMENT 13 As the name of his model suggests, Brown hypothesized that there was a lag in the response of consumption to changes in income and he used a partial adjustment model.

PARTIAL ADJUSTMENT 14 Substituting for desired consumption, one obtains current consumption in terms of current income and previous consumption.

PARTIAL ADJUSTMENT (7.4)(4.2)(2.8)(4.8) 15 Brown fitted the model with aggregate Canadian data for the years 1926–1949, omitting the years 1942 – 1945, using a simultaneous equations estimation technique. The variables were measured in billions of Canadian dollars at constant prices. t statistics are in parentheses.

16 PARTIAL ADJUSTMENT The short-run marginal propensities to consume out of wage and nonwage income are 0.61 and 0.28, respectively. Note that the former is indeed larger than the latter. How would you test whether the difference is significant? (7.4)(4.2)(2.8)(4.8)

17 PARTIAL ADJUSTMENT The coefficient of lagged consumption literally implies that, if consumption in the previous year had been 1 billion dollars greater, consumption this year would have been 0.22 billion dollars greater. (7.4)(4.2)(2.8)(4.8)

18 PARTIAL ADJUSTMENT That is a bit clumsy. It is better to interpret it with reference to in the adjustment process. It implies that the speed of adjustment is 0.78, meaning that 0.78 of the difference between desired and actual consumption is eliminated in one year. (7.4)(4.2)(2.8)(4.8)

19 PARTIAL ADJUSTMENT With the speed of adjustment, we can derive the long-run propensities to consume. We do this by dividing the short-run propensities by. We find that the long-run propensity to consume out of wages is (7.4)(4.2)(2.8)(4.8)

20 PARTIAL ADJUSTMENT Similarly, the long-run propensity to consume nonwage income is Note that, in this example, there is not a great difference between the short-run and long-run propensities. That is because the speed of adjustment is rapid. (7.4)(4.2)(2.8)(4.8)

============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ C LGDPI LGPRHOUS LGHOUS(-1) ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================ 21 PARTIAL ADJUSTMENT Here is the result of a parallel logarithmic regression of expenditure on housing on DPI and relative price, using the Demand Functions data set.

22 PARTIAL ADJUSTMENT The short-run income elasticity is ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ C LGDPI LGPRHOUS LGHOUS(-1) ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================

23 PARTIAL ADJUSTMENT The short-run price elasticity is Both of these elasticities are very low. This is because housing is a good example of a category of expenditure with slow adjustment. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ C LGDPI LGPRHOUS LGHOUS(-1) ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================

24 PARTIAL ADJUSTMENT The adjustment rate implicit in the coefficient of LGHOUS(–1) is only People do not change their housing quickly in response to changes in income and price. If anything, the estimated rate seems a little high. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ C LGDPI LGPRHOUS LGHOUS(-1) ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================

25 PARTIAL ADJUSTMENT The long-run income elasticity is 0.97, not far off the income elasticity in the static model in the first sequence for this chapter, ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ C LGDPI LGPRHOUS LGHOUS(-1) ============================================================ long-run income elasticity

26 PARTIAL ADJUSTMENT The long run price elasticity is 0.40, again not far from the estimate in the static model, In this example the long-run elasticities are much greater than the short-run ones because the speed of adjustment is slow. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ C LGDPI LGPRHOUS LGHOUS(-1) ============================================================ long-run price elasticity

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 11.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics