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The impact of energy prices
The impact of energy prices A housing market analysis
Ahmed
S. Zaki and Hans R. Isakson
This paper investigates the effect of energy costs on the housing market response. As the effect of energy costs has not been specifically investigated before in the literature, both a linear and non-linear model were investigated. The choice of the appropriate model was determined using the Box-Cox transformation technique. The chosen model was then validated. The results reveal that in the Spokane, Washington, area, where energy costs are rekztively low, energy prices do not have a significant effect on market response. However, applying the same methodology to areas where energy costs are higher might produce different results. Keywords:
The housing market response to different housing characteristics has been thoroughly investigated.‘-6 However, the housing market response to energy costs has not been investigated specifically except by Grether and Mieszkowski’ in 1973 when energy costs were relatively stable and inexpensive. Since then the cost of energy has been, and is expected to continue, rising. This rising cost can make the cost of energy a significant characteristic in determining the market response. Further research that includes energy cost as a housing characteristic seems warranted. To investigate the energy cost characteristic, a quantifiable measure is needed. To decide upon a measure, we present the following rationale. The response of a rational consumer depends upon the retail price of energy and its expected rate of increase. For a given market area these factors will be the same for all houses. However, the amount of energy consumed may vary widely from one house to another. For example, the amount of insulation in the envelope of the structure, the area of the windows, and the amount of air infiltration represent common factors associated
Ahmed Zaki is with the Department of Business Administration, Simon Fraser University, Burnaby, BC, Canada V5A lS6. Hans R. lsakson is with the Department of Finance and Real Estate, University of Texas, Arlington, TX 76019, USA. Final manuscript
100
received 7 September
1982.
0140-9883/83/020100-05
Energy: Prices; Housing
with energy consumption. Yet these factors are not readily determinable by the typical home buyer. Thus, home buyers usually rely upon other surrogate and perhaps less accurate factors. Two energy related factors of a house that are conveniently identifiable by a home buyer are the source of energy and the type of heating equipment. Dividing the price of the former by the thermal efficiency of the latter provides an approximate estimate of the heating costs. This study investigates the relative importance of energy prices to home buyers. A hedonic price approach is used to examine the response of house buyers to the heating cost relative to the other characteristics of the house. The sample The characteristics and selling prices of 13 18 houses were collected from the Multiple Listing Service records of the Spokane Board of Realtors, 1 July-30 September 1978. By concentrating on a short time period, the effect of time on energy prices can be ignored, while limiting the study to a relatively small area renders the effects of accessibility and other factors related to public services, taxes, and neighbourhood effects irrelevant.’ The housing characteristics considered in this study were chosen after a thorough review of previous housing market studies?-‘4 Table 1 gives these characteristics (variables), their symbols, and summary statistics.
503.00
0 1983 Butterworth & Co (Publishers) Ltd
The impact of energy prices: A. S. Zaki and H. R. Isakson Table 1. Housing characteristicsend summary statistics.
Characteristics
Notation
Mean
Standard deviation
House size (ft2) Plot size (ft2) Number of bedrooms Number of bathrooms Number of floors Number of fireplaces Basement finished (%I Year built Capacity of garage (cars) Number of recreation rooms Number of car ports Price of house (8) Price of heat ($/thermIa Dummy variables: = 1 if house heated with gas = 0 otherwise = 1 if house heated with electricity = 0 otherwise
HS LTS NBR NBT NFL NFP PB YB G RR CP SP PHT
1 113.85 11899.49 3.02 2.82 1.29 1.25 0.85 1953.60 1.25 0.73 0.03 43 533.05 45.39
19 176.80 0.90 1.16 0.56 0.75 0.29 24.13 0.79 0.49 0.18 23 430.11 8.01
395.72
PHT
1
0.0593
0.0768
PHT
2
0.2474
0.4318
Source: Thermal Performance for One and Two Family Dwellings, National Association of House Builders, Washington, DC, 1977. a The price of heat is defined as the marginal cost of the energy used to heat the house divided by the thermal efficiency of the heating equipment. All homes in the sample are heated by electricity, natural gas or fuel oil. The prices of these fuels (Q/therm) are 41.313, 12.394 and 30.171, respectively. The thermal efficiency of electricity, natural gas, and fuel oil are 1 .O, 0.60 and 0.50 respectively.
The sample is quite large and varied in the type of dwellings included. The representative (average) house has 1114 ft2 of living area on a 11900 ft2 plot with three bedrooms, nearly three bathrooms, a fireplace, a basement that is 85% complete, a one-car garage and a recreation room. The representative home was built in 1953 and sold for $43 533 during the summer of 1978. Dwellings, in the sample, range in size from 288-5621 ft2 of living area on one to four floors with one to eight bedrooms and one to eight bathrooms. The sample contains homes built as early as 188 1, while some homes are new; some homes have as many as five fireplaces and a five-car garage. The most expensive house in the sample sold for $250 000 while the least expensive sold for $5250. The model Multiple regression analysis is used to predict the price of a house P, as a function of the characteristic variables x, where x is an m-vector of the characteristic variables. p =f(x) Because it cannot be determined a priori whether f(x) is additive or multiplicative, both a linear and log-linear model are solved, and the choice of the appropriate model is determined using the transformation of Box and Cox.” Since the variables differ substantially in magnitude, the computation (X’X)-’ will suffer from serious roundoff errors. Any errors in (X’X)-’ may be magnified
ENERGY ECONOMICS
April 1983
when calculating the regression coefficients (B) or making other subsequent calculations.‘6 To overcome this problem, each observation Xii is standardized using the following transformation:
Xii- Xj dl(n-1) Sj
x+----
1
where Xij = the standardized ith observation variable ; n = the sample size ; ,?i = the mean of thejti variable; Sj = the standard deviation of thej”
of the iti
variable.
This transformation guarantees that all entries in the (X’X) matrix possess a mean of zero and a standard deviation of one so that the calculation of (X’X)-’ becomes much less subject to roundoff errors. Most statistical software packages include routines that perform this transformation. Additionally, normalization makes the comparison of the regression coefficients more meaningful. The original values of the regression coefficients (bi) are related to the transformed coefficients (I$-) by the fo!lowing formula: bi =
SP T
b;
3i where S, = the standard deviation of the dependent variable P. The stepwise procedure is used to select the best regression equation for both models. Draper and Smith” cite this method as the best of the variable selection procedures and recommend its use as long as a residual analysis is performed. Another study by Richardson and Thalheimer18 showed that the ‘all possible regression’ procedure did not yield better predictions than any of the other stepwise procedures. The stepwise procedure used in this study adds the variables to the model one by one if a variable’s F statistic is significant at the 0.5 level. However, after a variable is added, the procedure scans all other variables already in the model and deletes any variables that do not produce an F statistic significant at the 0.1 level. Thus, an independent variable may be dropped if it is no longer helpful in conjunction with the variables added at later stages. It is important to emphasize that the selection of the variable to be included or deleted from the model should not be viewed as hypothesis testing. Rather, it is a check up of computed F values to decide whether the variables have enough explanatory power to be included in the model. Finally, due to the nature of the independent variable of primary interest, the price of energy, the dummy variable technique is utilized to examine further the nature of the market’s reaction to the variable. Recall that each house in the sample is heated by either natural gas, electricity or fuel oil. Therefore, it is possible to collapse the variable PHT into a pair of dummy variables, PHT 1 and PHT 2 (PHT = 1 if house is heated by natural gas, 0 otherwise; PHT 2 =
101
The impact of energy prices: A. S. Zaki and H. R. Isakson house is heated by electricity, 0 otherwise.) Unfortunately, this procedure results in the loss of information concerning the magnitude of differences between the prices of natural gas, electricity and fuel oil. However, the procedure is useful in determining whether there are significant differences between the three types of heating having controlled for other factors that affect the price of a house.
ReSUltS The results of the last iteration of the stepwise procedure for both models are displayed in Table 2. The set of variables selected in the linear model differs from those in the log-linear model. For example, plot size is not significant in the log-linear model while the carport variable is insignificant in the linear model. To choose the appropriate functional form the Box-Cox transformation is applied to both models. The general form of this transformation is: z” - 1 Z’ =h
(4)
where Z = the variable being transformed ln the set {P, X} and he [0, 11. Griliches19 and Zarembka” present a detailed discussion on the use and appropriateness of this technique. In this study, nine values for X ranging from 0.1 to 0.9 were used to transform the data using Equation (4). Both models were run for each value of X and the value ofR* calculated as shown in Table 3. The R* value for both models increases as X approaches zero, Table 2. Revolts of last itamtion of stapwisa
analysis.
Linarr modal Variable
Coafficient
Standard error
NBT NFP PB
2 161.90 2 802.84
486.77 464.40 473.75 434.09 346.6 1 537.40 448.53 402.42 374.42
G
1511.95 1 991.68
LTS HS YB NFL
1354.68 8 659.95 3 196.58 1 053.55
Intercept
44571.82
F = 202.01
R2 = 0.7519
Log-linear model NBT NFP PB G HS YB NFL NBR CP
0.0591 0.1529
0.1031 0.0207
0.0193 0.0092
Intercept
10.643 R2=0.764
102
0.0068 0.0064 0.0067 0.0063 0.0069 0.0063 0.0053 0.0062 0.0054
0.0334 0.0648 0.0493
F = 470.34
Table 3. Results of applying Box-Cox
transformation.
R2 h
Linaar model
Log-linear model
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.7446 0.7445 0.7385 0.7283 0.7143 0.6965 0.6749 0.6494 0.6199
0.7548 0.7527 0.7463 0.7366 0.7239 0.7081 0.6892 0.6671 0.6417
indicating that the linear form is more appropriate than the log-linear form. Therefore, the linear model is selected for validation. Replacing PHT with the dummy variables PHT 1 and PHT 2 yields results identical to those reported above. That is, the dummy variables do not enter either the linear or the log-linear regression models. As a final check, both the linear and log-linear models are reestimated forcing first PHT and then PHT 1 and PHT 2 into the regression. In each case the t-statistic for the forced variable indicated that it does not have enough explanatory power while the coefficients of the other independent variables remain virtually unchanged.
Model validation The use of the stepwise regression analysis procedure requires the resultant model to be validated. In this study, model validation is achieved by examining the residuals of the model for anomalies and by testing for multicollinearity. Plots of the residuals against the independent and the dependent variables, as suggested by Draper and Smith*’ and Neter and Wassermans2 produce narrow bands of observations, indicating no anomalies. If these plots were to produce some other sort of pattern then it would have been necessary to adjust some or all of the independent or dependent variables in a manner similar to that used by Grether and Miezkowski.23 For example, if anomalies are associated with the house size variable, then dividing every other variable by house size frequently eliminates the problem. Often, the procedure to remove anomalies is one of trial and error in which several adjustments are made, and the resultant plots examined. The adjustment that produces the narrowest band of observations is selected as the best adjustment. Fortunately, adjustments of this nature were not required in this study. Another more systematic means of examining the residuals of the model is to calculate the unit normal deviate form of each residual ej using Vi = ei/Sp
i=l,2
,._.,
n.
(5)
If 95% of the values of V, lie between the limits (-1.96, 1.96) then the assumption that the error term of the regression is distributed normally with zero mean
ENERGY
ECONOMICS
April
1983
The impact of energy prices: A. S. Zaki and H. R. Isakson
and a standard deviation of one is correct.24 In this study 98% of the unit normal deviate form of the residuals falls within this interval indicating that the normality assumption is valid. Multicollinearity constitutes a threat to the proper specification, and the effective estimation of the structural relationship is found using the stepwise regression technique. Given substantial multicollinearity, the subsequent increase in sample standard errors virtually assures a tendency for relevant variables to be discarded incorrectly from regression equations. Farrar and Glauber25 and Haitovsky26 define multicollinearity in terms of departures from orthogonality in an independent variable set; ie as X’X approaches singularity then X’X approaches zero. Multicollinearity will almost always exist in nontrivial multiple regression models. Klein27 states, ‘ . . . intercorrelation
The most plausible explanation for this counter intuitive result is the relatively low price of energy in the Spokane area. In some other market areas the results might be very different. Also, as real energy prices continue to rise, the reaction of Spokane home buyers may very well change, making the price of heating an important factor to home buyers. Further study in this area is needed. The statistical procedures (normalization, stepwise regression analysis, the Box-Cox test and model validation) used in this study can easily be replicated elsewhere. Multiple listing services are found in most cities, and most computerized statistical packages easily perform the necessary statistical calculations. Finally, the statistical procedures employed in this study are useful to all others who wish to examine the importance of a particular real estate market phenomenon.
_. .‘. Thus, it is preferable to think of multicollinearity in terms of its severity, rather than its existence or non-existence. Haitovsky,28 based on a previous study by Farrar and Glauber,2a developed the following heuristic test:
References adverse selection and 1 P. T. Chinloy, ‘Depreciation, housing markets’, Journal of Urban Economics,
Vo15, 1978, pp 172-187. 2 P. F. Colwell and K. W. Foley. ‘Electric transmission lines and the selling price of residential property’, Appmisal Journal, Vo166, 1979, pp
He: X’X=O
490-499 .__ .__. 3 A. C. Goodman,
&:X’X#O x2(u) = k log (1 -
(6)
where
4
= (the number of degrees = im(m k = -[n 1- 5)] = the number variables considered (X’X) the matrix of simple correlation coefficients for In Equation (6), log below) X’X x2 indicates the
- X’X) is singularity.
A
zero (from value
H,, is accepted. Applying the above test to the model yields a x2 value of 114.5 which is significant only at less than the 0.0001 level. This result indicates that the estimates are not disturbed by multicollinearity.
5 6
7
; 10 11 :3 :I:
Summary and conclusion
VOl B26, 1964, pp 21 l-243.
The study reveals that energy prices do not have a significant effect on the housing market response for the summer of 1978 in the Spokane, Washington area. The analysis suggests that other factors, such as house size, plot size, year built, number of bedrooms, bathrooms, floors, and fireplaces, garage size and the percent of basement finished, are more important to home buyers than is the price of beat delivered into the house. Indeed, although the linear regression model is found to be more
appropriate than the log-linear form, the price of heat delivered into the house is shown to be insignificant in both forms.
ENERGY ECONOMICS
‘Hedonic prices, price indexes and housing markets’, Journal of Urban Economics, Vol5, 1978, pp 471-484. K. L. Guntermann, ‘FHA mortgage discount points, house prices, and consumer behavior’, AREUE Journal, Vol 7, 1979, pp 163-176. J. F. McDonald, ‘The use of proxy variables in housing price analysis’, Journal of Urban Economics, Vo17, 1980, pp 75-83. P. Mieszkowsky and A. M. Saper, ‘An estimate of the effects of airport noise on property values’, Journal of Urban Economics, Vo15, 1978, pp 425440. D. M. Grether and P. Mieszkowski, ‘Determinants of real estate values’, Journal of Urban Economics, Vol 1, 1974, pp 127-146. Ibid. Op tit, Ref 6. Op tit, Ref 4. Op tit, Ref 2. Op tit, Ref 3. Op tit, Ref 5. Op tit, Ref 1. G. E. P. Box and D. R. Cox, ‘An analysis of transformations’. Journal of the Royal Statistical Society._ ,
April 1983
_
J. Neter and W. Wasserman, Applied Linear Statistical Models, Richard D. Irwin, Homewood, IL, 1974. 17 N. Draper and Harry Smith, Applied Regression Analysis, John Wiley, New York, 1966. and R. Thalheimer, ‘Alternative 18 D. H. Richardson methods of variable selection: an application to real estate assessment’. AREUE Journal. Vol 7. 1979. pp 393-409. ’ 19 Z. Griliches, ‘Hedonic price indexes revisited’, in Z. Griliches, ed, Price Indexes and Quality Changes, Harvard University Press, Cambridge, MA, 197 1. of variables in 20 P. Zarembka. ‘Transformations economics’, in P. Zarembka, ed, Frontiers in Econometrics, Academic Press, New York, 1974. 21 Op tit, Ref 17. 16
103
The impact of energy prices: A. S. Zaki and H. R. Isakson ii 24 25
Op tit, Ref 16. Op tit, Ref 9. F. J. Anscombe and J. W. Tukey, ‘The examination and analysis of residuals’, Technometrics, Vol 5, 1963, pp 141-160. D. E. Farrar and R. R. Glauber, ‘Multicollinearity in regression analysis: the problem revisited’, Review of Economics and Statistics, Vo149, 1967, pp 92-
104
26 27
107. Y. Haitovsky, ‘Multicollinearity in regression analysis: comment’, Review of Economics and Statistics, Vo15 1, 1969, pp 486- 489. L. R. Klein, An Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, NJ, 1962. Op tit, Ref 26. Op tit, Ref 25.
ENERGY ECONOMICS
April 1983
Ahmed
S. Zaki and Hans R. Isakson
This paper investigates the effect of energy costs on the housing market response. As the effect of energy costs has not been specifically investigated before in the literature, both a linear and non-linear model were investigated. The choice of the appropriate model was determined using the Box-Cox transformation technique. The chosen model was then validated. The results reveal that in the Spokane, Washington, area, where energy costs are rekztively low, energy prices do not have a significant effect on market response. However, applying the same methodology to areas where energy costs are higher might produce different results. Keywords:
The housing market response to different housing characteristics has been thoroughly investigated.‘-6 However, the housing market response to energy costs has not been investigated specifically except by Grether and Mieszkowski’ in 1973 when energy costs were relatively stable and inexpensive. Since then the cost of energy has been, and is expected to continue, rising. This rising cost can make the cost of energy a significant characteristic in determining the market response. Further research that includes energy cost as a housing characteristic seems warranted. To investigate the energy cost characteristic, a quantifiable measure is needed. To decide upon a measure, we present the following rationale. The response of a rational consumer depends upon the retail price of energy and its expected rate of increase. For a given market area these factors will be the same for all houses. However, the amount of energy consumed may vary widely from one house to another. For example, the amount of insulation in the envelope of the structure, the area of the windows, and the amount of air infiltration represent common factors associated
Ahmed Zaki is with the Department of Business Administration, Simon Fraser University, Burnaby, BC, Canada V5A lS6. Hans R. lsakson is with the Department of Finance and Real Estate, University of Texas, Arlington, TX 76019, USA. Final manuscript
100
received 7 September
1982.
0140-9883/83/020100-05
Energy: Prices; Housing
with energy consumption. Yet these factors are not readily determinable by the typical home buyer. Thus, home buyers usually rely upon other surrogate and perhaps less accurate factors. Two energy related factors of a house that are conveniently identifiable by a home buyer are the source of energy and the type of heating equipment. Dividing the price of the former by the thermal efficiency of the latter provides an approximate estimate of the heating costs. This study investigates the relative importance of energy prices to home buyers. A hedonic price approach is used to examine the response of house buyers to the heating cost relative to the other characteristics of the house. The sample The characteristics and selling prices of 13 18 houses were collected from the Multiple Listing Service records of the Spokane Board of Realtors, 1 July-30 September 1978. By concentrating on a short time period, the effect of time on energy prices can be ignored, while limiting the study to a relatively small area renders the effects of accessibility and other factors related to public services, taxes, and neighbourhood effects irrelevant.’ The housing characteristics considered in this study were chosen after a thorough review of previous housing market studies?-‘4 Table 1 gives these characteristics (variables), their symbols, and summary statistics.
503.00
0 1983 Butterworth & Co (Publishers) Ltd
The impact of energy prices: A. S. Zaki and H. R. Isakson Table 1. Housing characteristicsend summary statistics.
Characteristics
Notation
Mean
Standard deviation
House size (ft2) Plot size (ft2) Number of bedrooms Number of bathrooms Number of floors Number of fireplaces Basement finished (%I Year built Capacity of garage (cars) Number of recreation rooms Number of car ports Price of house (8) Price of heat ($/thermIa Dummy variables: = 1 if house heated with gas = 0 otherwise = 1 if house heated with electricity = 0 otherwise
HS LTS NBR NBT NFL NFP PB YB G RR CP SP PHT
1 113.85 11899.49 3.02 2.82 1.29 1.25 0.85 1953.60 1.25 0.73 0.03 43 533.05 45.39
19 176.80 0.90 1.16 0.56 0.75 0.29 24.13 0.79 0.49 0.18 23 430.11 8.01
395.72
PHT
1
0.0593
0.0768
PHT
2
0.2474
0.4318
Source: Thermal Performance for One and Two Family Dwellings, National Association of House Builders, Washington, DC, 1977. a The price of heat is defined as the marginal cost of the energy used to heat the house divided by the thermal efficiency of the heating equipment. All homes in the sample are heated by electricity, natural gas or fuel oil. The prices of these fuels (Q/therm) are 41.313, 12.394 and 30.171, respectively. The thermal efficiency of electricity, natural gas, and fuel oil are 1 .O, 0.60 and 0.50 respectively.
The sample is quite large and varied in the type of dwellings included. The representative (average) house has 1114 ft2 of living area on a 11900 ft2 plot with three bedrooms, nearly three bathrooms, a fireplace, a basement that is 85% complete, a one-car garage and a recreation room. The representative home was built in 1953 and sold for $43 533 during the summer of 1978. Dwellings, in the sample, range in size from 288-5621 ft2 of living area on one to four floors with one to eight bedrooms and one to eight bathrooms. The sample contains homes built as early as 188 1, while some homes are new; some homes have as many as five fireplaces and a five-car garage. The most expensive house in the sample sold for $250 000 while the least expensive sold for $5250. The model Multiple regression analysis is used to predict the price of a house P, as a function of the characteristic variables x, where x is an m-vector of the characteristic variables. p =f(x) Because it cannot be determined a priori whether f(x) is additive or multiplicative, both a linear and log-linear model are solved, and the choice of the appropriate model is determined using the transformation of Box and Cox.” Since the variables differ substantially in magnitude, the computation (X’X)-’ will suffer from serious roundoff errors. Any errors in (X’X)-’ may be magnified
ENERGY ECONOMICS
April 1983
when calculating the regression coefficients (B) or making other subsequent calculations.‘6 To overcome this problem, each observation Xii is standardized using the following transformation:
Xii- Xj dl(n-1) Sj
x+----
1
where Xij = the standardized ith observation variable ; n = the sample size ; ,?i = the mean of thejti variable; Sj = the standard deviation of thej”
of the iti
variable.
This transformation guarantees that all entries in the (X’X) matrix possess a mean of zero and a standard deviation of one so that the calculation of (X’X)-’ becomes much less subject to roundoff errors. Most statistical software packages include routines that perform this transformation. Additionally, normalization makes the comparison of the regression coefficients more meaningful. The original values of the regression coefficients (bi) are related to the transformed coefficients (I$-) by the fo!lowing formula: bi =
SP T
b;
3i where S, = the standard deviation of the dependent variable P. The stepwise procedure is used to select the best regression equation for both models. Draper and Smith” cite this method as the best of the variable selection procedures and recommend its use as long as a residual analysis is performed. Another study by Richardson and Thalheimer18 showed that the ‘all possible regression’ procedure did not yield better predictions than any of the other stepwise procedures. The stepwise procedure used in this study adds the variables to the model one by one if a variable’s F statistic is significant at the 0.5 level. However, after a variable is added, the procedure scans all other variables already in the model and deletes any variables that do not produce an F statistic significant at the 0.1 level. Thus, an independent variable may be dropped if it is no longer helpful in conjunction with the variables added at later stages. It is important to emphasize that the selection of the variable to be included or deleted from the model should not be viewed as hypothesis testing. Rather, it is a check up of computed F values to decide whether the variables have enough explanatory power to be included in the model. Finally, due to the nature of the independent variable of primary interest, the price of energy, the dummy variable technique is utilized to examine further the nature of the market’s reaction to the variable. Recall that each house in the sample is heated by either natural gas, electricity or fuel oil. Therefore, it is possible to collapse the variable PHT into a pair of dummy variables, PHT 1 and PHT 2 (PHT = 1 if house is heated by natural gas, 0 otherwise; PHT 2 =
101
The impact of energy prices: A. S. Zaki and H. R. Isakson house is heated by electricity, 0 otherwise.) Unfortunately, this procedure results in the loss of information concerning the magnitude of differences between the prices of natural gas, electricity and fuel oil. However, the procedure is useful in determining whether there are significant differences between the three types of heating having controlled for other factors that affect the price of a house.
ReSUltS The results of the last iteration of the stepwise procedure for both models are displayed in Table 2. The set of variables selected in the linear model differs from those in the log-linear model. For example, plot size is not significant in the log-linear model while the carport variable is insignificant in the linear model. To choose the appropriate functional form the Box-Cox transformation is applied to both models. The general form of this transformation is: z” - 1 Z’ =h
(4)
where Z = the variable being transformed ln the set {P, X} and he [0, 11. Griliches19 and Zarembka” present a detailed discussion on the use and appropriateness of this technique. In this study, nine values for X ranging from 0.1 to 0.9 were used to transform the data using Equation (4). Both models were run for each value of X and the value ofR* calculated as shown in Table 3. The R* value for both models increases as X approaches zero, Table 2. Revolts of last itamtion of stapwisa
analysis.
Linarr modal Variable
Coafficient
Standard error
NBT NFP PB
2 161.90 2 802.84
486.77 464.40 473.75 434.09 346.6 1 537.40 448.53 402.42 374.42
G
1511.95 1 991.68
LTS HS YB NFL
1354.68 8 659.95 3 196.58 1 053.55
Intercept
44571.82
F = 202.01
R2 = 0.7519
Log-linear model NBT NFP PB G HS YB NFL NBR CP
0.0591 0.1529
0.1031 0.0207
0.0193 0.0092
Intercept
10.643 R2=0.764
102
0.0068 0.0064 0.0067 0.0063 0.0069 0.0063 0.0053 0.0062 0.0054
0.0334 0.0648 0.0493
F = 470.34
Table 3. Results of applying Box-Cox
transformation.
R2 h
Linaar model
Log-linear model
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.7446 0.7445 0.7385 0.7283 0.7143 0.6965 0.6749 0.6494 0.6199
0.7548 0.7527 0.7463 0.7366 0.7239 0.7081 0.6892 0.6671 0.6417
indicating that the linear form is more appropriate than the log-linear form. Therefore, the linear model is selected for validation. Replacing PHT with the dummy variables PHT 1 and PHT 2 yields results identical to those reported above. That is, the dummy variables do not enter either the linear or the log-linear regression models. As a final check, both the linear and log-linear models are reestimated forcing first PHT and then PHT 1 and PHT 2 into the regression. In each case the t-statistic for the forced variable indicated that it does not have enough explanatory power while the coefficients of the other independent variables remain virtually unchanged.
Model validation The use of the stepwise regression analysis procedure requires the resultant model to be validated. In this study, model validation is achieved by examining the residuals of the model for anomalies and by testing for multicollinearity. Plots of the residuals against the independent and the dependent variables, as suggested by Draper and Smith*’ and Neter and Wassermans2 produce narrow bands of observations, indicating no anomalies. If these plots were to produce some other sort of pattern then it would have been necessary to adjust some or all of the independent or dependent variables in a manner similar to that used by Grether and Miezkowski.23 For example, if anomalies are associated with the house size variable, then dividing every other variable by house size frequently eliminates the problem. Often, the procedure to remove anomalies is one of trial and error in which several adjustments are made, and the resultant plots examined. The adjustment that produces the narrowest band of observations is selected as the best adjustment. Fortunately, adjustments of this nature were not required in this study. Another more systematic means of examining the residuals of the model is to calculate the unit normal deviate form of each residual ej using Vi = ei/Sp
i=l,2
,._.,
n.
(5)
If 95% of the values of V, lie between the limits (-1.96, 1.96) then the assumption that the error term of the regression is distributed normally with zero mean
ENERGY
ECONOMICS
April
1983
The impact of energy prices: A. S. Zaki and H. R. Isakson
and a standard deviation of one is correct.24 In this study 98% of the unit normal deviate form of the residuals falls within this interval indicating that the normality assumption is valid. Multicollinearity constitutes a threat to the proper specification, and the effective estimation of the structural relationship is found using the stepwise regression technique. Given substantial multicollinearity, the subsequent increase in sample standard errors virtually assures a tendency for relevant variables to be discarded incorrectly from regression equations. Farrar and Glauber25 and Haitovsky26 define multicollinearity in terms of departures from orthogonality in an independent variable set; ie as X’X approaches singularity then X’X approaches zero. Multicollinearity will almost always exist in nontrivial multiple regression models. Klein27 states, ‘ . . . intercorrelation
The most plausible explanation for this counter intuitive result is the relatively low price of energy in the Spokane area. In some other market areas the results might be very different. Also, as real energy prices continue to rise, the reaction of Spokane home buyers may very well change, making the price of heating an important factor to home buyers. Further study in this area is needed. The statistical procedures (normalization, stepwise regression analysis, the Box-Cox test and model validation) used in this study can easily be replicated elsewhere. Multiple listing services are found in most cities, and most computerized statistical packages easily perform the necessary statistical calculations. Finally, the statistical procedures employed in this study are useful to all others who wish to examine the importance of a particular real estate market phenomenon.
_. .‘. Thus, it is preferable to think of multicollinearity in terms of its severity, rather than its existence or non-existence. Haitovsky,28 based on a previous study by Farrar and Glauber,2a developed the following heuristic test:
References adverse selection and 1 P. T. Chinloy, ‘Depreciation, housing markets’, Journal of Urban Economics,
Vo15, 1978, pp 172-187. 2 P. F. Colwell and K. W. Foley. ‘Electric transmission lines and the selling price of residential property’, Appmisal Journal, Vo166, 1979, pp
He: X’X=O
490-499 .__ .__. 3 A. C. Goodman,
&:X’X#O x2(u) = k log (1 -
(6)
where
4
= (the number of degrees = im(m k = -[n 1- 5)] = the number variables considered (X’X) the matrix of simple correlation coefficients for In Equation (6), log below) X’X x2 indicates the
- X’X) is singularity.
A
zero (from value
H,, is accepted. Applying the above test to the model yields a x2 value of 114.5 which is significant only at less than the 0.0001 level. This result indicates that the estimates are not disturbed by multicollinearity.
5 6
7
; 10 11 :3 :I:
Summary and conclusion
VOl B26, 1964, pp 21 l-243.
The study reveals that energy prices do not have a significant effect on the housing market response for the summer of 1978 in the Spokane, Washington area. The analysis suggests that other factors, such as house size, plot size, year built, number of bedrooms, bathrooms, floors, and fireplaces, garage size and the percent of basement finished, are more important to home buyers than is the price of beat delivered into the house. Indeed, although the linear regression model is found to be more
appropriate than the log-linear form, the price of heat delivered into the house is shown to be insignificant in both forms.
ENERGY ECONOMICS
‘Hedonic prices, price indexes and housing markets’, Journal of Urban Economics, Vol5, 1978, pp 471-484. K. L. Guntermann, ‘FHA mortgage discount points, house prices, and consumer behavior’, AREUE Journal, Vol 7, 1979, pp 163-176. J. F. McDonald, ‘The use of proxy variables in housing price analysis’, Journal of Urban Economics, Vo17, 1980, pp 75-83. P. Mieszkowsky and A. M. Saper, ‘An estimate of the effects of airport noise on property values’, Journal of Urban Economics, Vo15, 1978, pp 425440. D. M. Grether and P. Mieszkowski, ‘Determinants of real estate values’, Journal of Urban Economics, Vol 1, 1974, pp 127-146. Ibid. Op tit, Ref 6. Op tit, Ref 4. Op tit, Ref 2. Op tit, Ref 3. Op tit, Ref 5. Op tit, Ref 1. G. E. P. Box and D. R. Cox, ‘An analysis of transformations’. Journal of the Royal Statistical Society._ ,
April 1983
_
J. Neter and W. Wasserman, Applied Linear Statistical Models, Richard D. Irwin, Homewood, IL, 1974. 17 N. Draper and Harry Smith, Applied Regression Analysis, John Wiley, New York, 1966. and R. Thalheimer, ‘Alternative 18 D. H. Richardson methods of variable selection: an application to real estate assessment’. AREUE Journal. Vol 7. 1979. pp 393-409. ’ 19 Z. Griliches, ‘Hedonic price indexes revisited’, in Z. Griliches, ed, Price Indexes and Quality Changes, Harvard University Press, Cambridge, MA, 197 1. of variables in 20 P. Zarembka. ‘Transformations economics’, in P. Zarembka, ed, Frontiers in Econometrics, Academic Press, New York, 1974. 21 Op tit, Ref 17. 16
103
The impact of energy prices: A. S. Zaki and H. R. Isakson ii 24 25
Op tit, Ref 16. Op tit, Ref 9. F. J. Anscombe and J. W. Tukey, ‘The examination and analysis of residuals’, Technometrics, Vol 5, 1963, pp 141-160. D. E. Farrar and R. R. Glauber, ‘Multicollinearity in regression analysis: the problem revisited’, Review of Economics and Statistics, Vo149, 1967, pp 92-
104
26 27
107. Y. Haitovsky, ‘Multicollinearity in regression analysis: comment’, Review of Economics and Statistics, Vo15 1, 1969, pp 486- 489. L. R. Klein, An Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, NJ, 1962. Op tit, Ref 26. Op tit, Ref 25.
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