HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std.
Download ReportTranscript HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std.
HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.005625 0.167903 0.033501 0.9734 LGDPI 1.031918 0.006649 155.1976 0.0000 LGPRHOUS -0.483421 0.041780 -11.57056 0.0000 ============================================================ R-squared 0.998583 Mean dependent var 6.359334 Adjusted R-squared 0.998515 S.D. dependent var 0.437527 S.E. of regression 0.016859 Akaike info criter-5.263574 Sum squared resid 0.011937 Schwarz criterion -5.143130 Log likelihood 121.4304 F-statistic 14797.05 Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000 ============================================================ This sequence gives an example of how a direct examination of plots of the residuals and the data for the variables in a regression model may lead to an improvement in the specification of the regression model. 1 HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.005625 0.167903 0.033501 0.9734 LGDPI 1.031918 0.006649 155.1976 0.0000 LGPRHOUS -0.483421 0.041780 -11.57056 0.0000 ============================================================ R-squared 0.998583 Mean dependent var 6.359334 Adjusted R-squared 0.998515 S.D. dependent var 0.437527 S.E. of regression 0.016859 Akaike info criter-5.263574 Sum squared resid 0.011937 Schwarz criterion -5.143130 Log likelihood 121.4304 F-statistic 14797.05 Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000 ============================================================ The regression output is that for a logarithmic regression of aggregate expenditure on housing services on income and relative price for the United States for the period 1959– 2003. The income and price elasticities seem plausible. 2 HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Breusch–Godfrey statistic: 20.02 Method: Least Squares Sample: 1959 2003 critical value of c2(1), 0.1%, is 10.83 Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.005625 0.167903 0.033501 0.9734 LGDPI 1.031918 0.006649 155.1976 0.0000 LGPRHOUS -0.483421 0.041780 -11.57056 0.0000 ============================================================ R-squared 0.998583 Mean dependent var 6.359334 Adjusted R-squared 0.998515 S.D. dependent var 0.437527 S.E. of regression 0.016859 Akaike info criter-5.263574 Sum squared resid 0.011937 Schwarz criterion -5.143130 Log likelihood 121.4304 F-statistic 14797.05 Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000 ============================================================ However, the Breusch–Godfrey and Durbin–Watson statistics both indicate autocorrelation at a high significance level. 3 HOUSING DYNAMICS 0.04 0.03 0.02 0.01 0 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 -0.01 -0.02 -0.03 -0.04 The residuals exhibit a classic pattern of strong positive autocorrelation. 4 HOUSING DYNAMICS 9 0.04 0.03 8 0.02 0.01 7 0 6 -0.01 -0.02 5 -0.03 4 -0.04 1959 1963 1967 1971 LGHOUS 1975 FITTED 1979 1983 LGDPI 1987 1991 LGPRHOUS 1995 1999 2003 RESIDS The actual and fitted values of the dependent variable and the series for income and price have been added to the diagram. The price series was very flat and so had little influence on the fitted values. It will be ignored in the discussion that follows. 5 HOUSING DYNAMICS 9 0.04 0.03 8 0.02 0.01 7 0 6 -0.01 -0.02 5 -0.03 4 -0.04 1959 1963 1967 1971 LGHOUS 1975 FITTED 1979 1983 LGDPI 1987 1991 LGPRHOUS 1995 1999 2003 RESIDS There was a very large negative residual in 1973. We will enlarge this part of the diagram and take a closer look. 6 HOUSING DYNAMICS 6.4 8.2 6.3 8.1 6.2 8 6.1 6 7.9 1971 1972 1973 LGHOUS FITTED 1974 1975 LGDPI In 1973, income (right scale) grew unusually rapidly. The fitted value of housing expenditure (left scale, with actual value) accordingly rose above its trend. 7 HOUSING DYNAMICS 6.4 8.2 6.3 8.1 6.2 8 6.1 6 7.9 1971 1972 1973 LGHOUS FITTED 1974 1975 LGDPI This boom was stopped in its tracks by the first oil shock. Income actually declined in 1974, the only fall in the entire sample period. 8 HOUSING DYNAMICS 6.4 8.2 6.3 8.1 6.2 8 6.1 6 7.9 1971 1972 1973 LGHOUS FITTED 1974 1975 LGDPI As a consequence, the fitted value of housing expenditure would also have fallen in 1974. In actual fact it rose a little because the real price of housing fell relatively sharply in 1974. 9 HOUSING DYNAMICS 6.4 8.2 6.3 8.1 6.2 8 6.1 6 7.9 1971 1972 1973 LGHOUS FITTED 1974 1975 LGDPI However, the actual value of housing maintained its previous trend in those two years, responding not at all to the short-run variations in the growth of income. This accounts for the gap that opened up in 1973, and the large negative residual in that year. 10 HOUSING DYNAMICS 9 0.04 0.03 8 0.02 0.01 7 0 6 -0.01 -0.02 5 -0.03 4 -0.04 1959 1963 1967 1971 LGHOUS 1975 FITTED 1979 1983 LGDPI 1987 1991 LGPRHOUS 1995 1999 2003 RESIDS There was a similar large negative residual in 1984. We will enlarge this part of the diagram. 11 HOUSING DYNAMICS 8.5 6.6 8.4 6.5 8.3 8.2 6.4 1982 1983 1984 LGHOUS 1986 1985 FITTED 1987 LGDPI Income grew unusually rapidly in 1984. As a consequence, the fitted value of housing also grew rapidly. However the actual value of housing grew at much the same rate as previously. Hence the negative residual. 12 HOUSING DYNAMICS 8.5 6.6 8.4 6.5 8.3 8.2 6.4 1982 1983 1984 LGHOUS 1986 1985 FITTED 1987 LGDPI In the years immediately after 1984, income grew at a slower rate. Accordingly the fitted value of housing grew at a slower rate. But the actual value of housing grew at much the same rate as before, turning the negative residual in 1984 into a large positive one in 1987. 13 HOUSING DYNAMICS 9 0.04 0.03 8 0.02 0.01 7 0 6 -0.01 -0.02 5 -0.03 4 -0.04 1959 1963 1967 1971 LGHOUS 1975 FITTED 1979 1983 LGDPI 1987 1991 LGPRHOUS 1995 1999 2003 RESIDS Finally, we shall take a closer look at the series of positive residuals from 1960 to 1965. 14 HOUSING DYNAMICS 5.9 7.8 7.7 5.8 7.6 5.7 7.5 5.6 7.4 5.5 7.3 5.4 7.2 1959 1960 1961 1962 LGHOUS 1963 FITTED 1964 1965 1966 LGDPI In the first part of this subperiod, income was growing relatively slowly. Towards the end, it started to accelerate. The fitted values followed suit. 15 HOUSING DYNAMICS 5.9 7.8 7.7 5.8 7.6 5.7 7.5 5.6 7.4 5.5 7.3 5.4 7.2 1959 1960 1961 1962 LGHOUS 1963 FITTED 1964 1965 1966 LGDPI However, the actual values maintained a constant trend. Because it was unresponsive to the variations in the growth rate of income, a gap opened up in the middle, giving rise to the positive residuals. 16 HOUSING DYNAMICS 5.9 7.8 7.7 5.8 7.6 5.7 7.5 5.6 7.4 5.5 7.3 5.4 7.2 1959 1960 1961 1962 LGHOUS 1963 FITTED 1964 1965 1966 LGDPI In this case, as in the previous two, the residuals are not being caused by autocorrelation. If that were the case, the actual values should be relatively volatile, compared with the trend of the fitted values. 17 HOUSING DYNAMICS 5.9 7.8 7.7 5.8 7.6 5.7 7.5 5.6 7.4 5.5 7.3 5.4 7.2 1959 1960 1961 1962 LGHOUS 1963 FITTED 1964 1965 1966 LGDPI What we see here is exactly the opposite. The actual values have a very stable trend, while the fitted values respond, as they must, to short-run variations in the growth of income. The pattern we see in the residuals is caused by the nonresponse of the actual values. 18 HOUSING DYNAMICS 5.9 7.8 7.7 5.8 7.6 5.7 7.5 5.6 7.4 5.5 7.3 5.4 7.2 1959 1960 1961 1962 LGHOUS 1963 FITTED 1964 1965 1966 LGDPI One way to model the inertia in the growth rate of the actual values is to add a lagged dependent variable to the regression model. 19 HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.073957 0.062915 1.175499 0.2467 LGDPI 0.282935 0.046912 6.031246 0.0000 LGPRHOUS -0.116949 0.027383 -4.270880 0.0001 LGHOUS(-1) 0.707242 0.044405 15.92699 0.0000 ============================================================ R-squared 0.999795 Mean dependent var 6.379059 Adjusted R-squared 0.999780 S.D. dependent var 0.421861 S.E. of regression 0.006257 Akaike info criter-7.223711 Sum squared resid 0.001566 Schwarz criterion -7.061512 Log likelihood 162.9216 F-statistic 65141.75 Durbin-Watson stat 1.810958 Prob(F-statistic) 0.000000 ===================================================================== We are now hypothesizing that current expenditure on housing services depends on previous expenditure as well as income and price. Here is the regression with the lagged dependent variable added to the model. 20 HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.073957 0.062915 1.175499 0.2467 LGDPI 0.282935 0.046912 6.031246 0.0000 LGPRHOUS -0.116949 0.027383 -4.270880 0.0001 LGHOUS(-1) 0.707242 0.044405 15.92699 0.0000 ============================================================ R-squared 0.999795 Mean dependent var 6.379059 Adjusted R-squared 0.999780 S.D. dependent var 0.421861 S.E. of regression 0.006257 Akaike info criter-7.223711 Sum squared resid 0.001566 Schwarz criterion -7.061512 Log likelihood 162.9216 F-statistic 65141.75 Durbin-Watson stat 1.810958 Prob(F-statistic) 0.000000 ===================================================================== The Durbin–Watson statistic, previously 0.63, is now quite close to 2. Of course, since we have a lagged dependent variable in the model, we should look at the h statistic instead. 21 HOUSING DYNAMICS 44 ============================================================ h 0.095 0.66 Dependent Variable: LGHOUS 1 44 0.0020 Method: Least Squares Sample(adjusted): 1960 2003 Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.073957 0.062915 1.175499 0.2467 LGDPI 0.282935 0.046912 6.031246 0.0000 LGPRHOUS -0.116949 0.027383 -4.270880 0.0001 LGHOUS(-1) 0.707242 0.044405 15.92699 0.0000 ============================================================ R-squared 0.999795 Mean dependent var 6.379059 Adjusted R-squared 0.999780 S.D. dependent var 0.421861 S.E. of regression 0.006257 Akaike info criter-7.223711 Sum squared resid 0.001566 Schwarz criterion -7.061512 Log likelihood 162.9216 F-statistic 65141.75 Durbin-Watson stat 1.810958 Prob(F-statistic) 0.000000 ===================================================================== We calculated the h statistic for this regression in the previous sequence. It is 0.66, and so now we do not reject the null hypothesis of no autocorrelation at the 5% significance level (critical value 1.96). Strictly speaking, of course, the test is valid only in large samples. 22 HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.073957 0.062915 1.175499 0.2467 LGDPI 0.282935 0.046912 6.031246 0.0000 LGPRHOUS -0.116949 0.027383 -4.270880 0.0001 LGHOUS(-1) 0.707242 0.044405 15.92699 0.0000 ============================================================ R-squared 0.999795 Mean dependent var 6.379059 Adjusted R-squared 0.999780 S.D. dependent var 0.421861 S.E. of regression 0.006257 Akaike info criter-7.223711 Sum squared resid 0.001566 Schwarz criterion -7.061512 Log likelihood 162.9216 F-statistic 65141.75 Durbin-Watson stat 1.810958 Prob(F-statistic) 0.000000 ============================================================ The new equation indicates that current expenditure on housing services is determined only partly by current income and price. Previous expenditure is clearly very important as well. 23 HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.005625 0.167903 0.033501 0.9734 LGDPI 1.031918 0.006649 155.1976 0.0000 LGPRHOUS -0.483421 0.041780 -11.57056 0.0000 ============================================================ Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000 ============================================================ ============================================================ Dependent Variable: LGHOUS ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.073957 0.062915 1.175499 0.2467 LGDPI 0.282935 0.046912 6.031246 0.0000 LGPRHOUS -0.116949 0.027383 -4.270880 0.0001 LGHOUS(-1) 0.707242 0.044405 15.92699 0.0000 ============================================================ Durbin-Watson stat 1.810958 Prob(F-statistic) 0.000000 ============================================================ The apparent autocorrelation exhibited by the residuals in the plot, and the resulting low value of the d statistic in the original regression, were thus attributable to the omission of an important variable, rather than to the disturbance term being subject to an AR(1) process. 24 HOUSING DYNAMICS ============================================================ Dependent Variable: LGHOUS ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.005625 0.167903 0.033501 0.9734 LGDPI 1.031918 0.006649 155.1976 0.0000 LGPRHOUS -0.483421 0.041780 -11.57056 0.0000 ============================================================ Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000 ============================================================ ============================================================ Dependent Variable: LGHOUS ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.073957 0.062915 1.175499 0.2467 LGDPI 0.282935 0.046912 6.031246 0.0000 LGPRHOUS -0.116949 0.027383 -4.270880 0.0001 LGHOUS(-1) 0.707242 0.044405 15.92699 0.0000 ============================================================ Durbin-Watson stat 1.810958 Prob(F-statistic) 0.000000 ============================================================ Note that the income and price elasticities are much lower than in the original regression. We have already seen the reason for this in the sequence that discussed the dynamics inherent in a partial adjustment model. 25 Copyright Christopher Dougherty 2011. 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 12.4 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 http://www.oup.com/uk/orc/bin/9780199567089/. 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 http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lse. 11.07.25