The purpose of this article is to build a residential investment model for Sacramento County. The population increased by 54.4% while housing units increased by 64.3% during the same period of 1980 to 2000. The importance of the housing sector to the economy derives from the fact that one-third of all private gross investment in the U.S.A. is in the residential sector of the economy according to Bonnie (1998). Housing investment represents 4% of the GDP according to Sieders (1997). An additional 8% of the GDP consists of housing services to owners and renters in the form of personal expenditures, and another 8% of GDP represents consumer spending on housing related goods and services such as furniture, utilities, appliances, etc. The sum of the above expenditures used directly or indirectly on housing represents 20% of the GDP. Therefore, the impact of a change in spending on housing will lead to a recession or expansion of the GDP. The significant size of this total investment deserves an investigation. The model will show the quantitative impact of credit condition, construction costs, income, population, housing stock, and other factors on housing industry.

Two important factors are responsible for an increase in the demand for housing. The rapid increase in income of the baby boomer households during the last 20 years and the creation of adjustable mortgages (ARM) created a strong demand for housing. The creative funding through mortgage-backed securities and through Rent-To-Own (RTO) increased the investment in housing.

According to Dipasquale (1999), the housing supply is not only impacted by new units exclusively, but also by the decisions of the owners of existing housing stock. Existing housing can be converted from large single-family dwellings into several small apartments; conversely, two or more apartments can be combined to form a single unit. Owners can also repair existing units or decrease maintenance. Government policy influences the supply side through construction of public housing. Tax policy is used to encourage or discourage private investment in new rental housing. Government regulations involving environment and growth issues must be factored in. The Economic Recovery Tax Act of 1981 encouraged investment in housing while the Tax Reform Act of 1986 decreased the favorable tax treatment of rental housing. The Tax Reform Act increased the capital gains rate, changed the accelerated depreciation for tax purposes to straight-line depreciation, and increased the tax life from the previous limit of 19 years to 27.5 years.

Construction Costs: According to Somerville (1999) the cost of material represents 65% of the total cost of building residential homes, while the remaining 35% represents labor cost. A significant number of empirical literature on housing construction failed to show a negative relationship between housing starts and the cost of construction. As a matter of fact, the empirical literature demonstrates the opposite result (see Potpan, 1996, Poterba 1984, and Topel and Rosen 1988). Somerville (1999) on the other hand, showed a negative relationship between housing starts and construction cost using new quality control construction costs based on micro data in three metropolitan areas. The impact of mortgage rates on the supply of residential housing will be explored later in the paper.

Housing demand is influenced by many variables. These variables include housing finance, mortgage rates, demographic trends, advances in technology, future capital gains, and disposable income.

Housing Finance: The wide spread securitization of housing mortgages in the 1980's facilitated the financing of residential demand. According to Hu (1992), the mortgage-backed securities widened the investor base enabling new domestic and foreign investors to become funding sources. Also, Lereah (1997) argued that financing home purchases has shifted gradually from savings and loan associations to mortgage banking companies. This shift was demonstrated by the increase in the mortgage banking companies= market share of mortgage originations from 35% in 1990 to 56% by 1996.

Mortgage Rate: The mortgage rate is another factor influencing the demand for housing. The higher the mortgage rate, the higher the monthly payments; therefore, fewer households will be eligible for mortgage finance. The strong demand for housing, mostly from the baby-boomers, and the creation of the adjustable rate mortgage (ARM) contributed significantly to the housing expansion in the 1980's. Low interest rate and a steady growth of income in the 1990's contributed to the continued expansion of the demand for residential housing. The fixed rate mortgages (FRM) are preferred to ARMs by consumers when the difference between the two rates is small. The ARM reduces the sensitivity of residential housing demand to interest rate changes. When the mortgage rate is high, homebuyers will use the ARM for finance. When the interest rate is low, homebuyers will use the FRM. Prepayment of mortgages will increase in a declining interest rate environment due to home sales or mortgage refinancing. As the spread between the ARM and FRM increases, during any given time, the homebuyers will use ARM for financing. The impact of the spread between the two rates was clear in 1996, when ARMs reached 26% of the market share of total originations of mortgages (see Lereah 1997 and see Plaut 1991). The increase of prepayment of mortgages by borrowers through refinancing of residential housing during the decline in the market mortgage rate represents a risk for the lender of FRMs.

This risk for the lenders is the result of reinvesting their money at a lower rate of return. But, if the interest rate increases, the borrower of the FRM will not be inclined to prepay the loan. The lender will suffer capital losses. Hu argued (1992) that the decline of the interest rate has to be at least 150 basis points below the mortgage rate to create economic incentives for borrowers to refinance. Otherwise, the high transaction cost of refinancing eliminates the benefit (to the homeowner) of refinancing mortgages. The creation of ARMs solved the lender's risk associated with mortgage lending. Another way of reducing the risk to homebuilders is their ability to sell the new house before construction starts. This high rate of turnover of new houses makes construction loans to homebuilders less risky and makes construction loans for new housing more easily found.

According to Chan (1999), most firms engaged in residential construction are small in size. Therefore, these firms face the denial of credit, or they end up paying a premium risk during the period of a decline in credit supply. This higher premium contributes to higher construction loan costs during tight credit conditions. Some builders will try to avoid paying this high risk premium by delaying or canceling their construction projects.

Demographic Trends: Demographic trends are another factor influencing the demand for housing. Baby boomers started buying homes during the 1980s and 1990s for the first time in large numbers. The two-income families have increased from 63% of households in 1980 to 80% by 1986, according to Lereah (1997). Population growth, the increase of immigration and migration, especially to California, combined with an increase of household formation rate contributed significantly to the increase in the demand for housing. Phillips (1991) showed that households in which the wife works as a professional or manager spend 17% more on the average on housing than households where the wife holds different occupations. This additional spending may be the result of higher expected permanent income level. Since housing is a normal good, we would expect a positive relationship between expected permanent income and expenditures on housing. Also, the opportunity cost of professional jobs is higher than for non-professional jobs. Therefore, one would expect the professional woman to work longer hours, resulting in a higher permanent income level for the family.

Makiw and Weil (1989) expected that the aging of the U.S. population and the decline of the 20 to 30 year age group would lead to a 40% decline in housing prices in real terms between 1990 and 2010. Their expectation may apply to other areas of the U.S., but there is no evidence that can be applied to California so far. On the contrary, we see the reverse is true. The rise of the duel professional household offsets any expected decline in housing demand due to the change in the population structure.

Technological Advancements and Other Variables: Advances in technology led to the acceleration of the loan approval process and the creation of large mortgage companies with the advantage of economies to scale. It is expected that the current and future disposable income, as well as the population, will have a positive impact on housing demand. But housing stocks are expected to have a negative impact on new housing units. The larger the housing stocks, especially unsold units, the smaller the incentive for the homebuilders to invest in new residential homes. The literature ignores the role of the land prices in determining housing investment. The exception is Dipasquale and Wheaton (1994) who found that land cost did not have a significant impact on housing investment.

All of the above combined with mortgage interest reduction, fair lending policies, and capital gains tax exclusions contributed to the growth of the demand and supply for housing.

The total number of housing units of 1990 – 2000 is taken from the 2000 Population and Housing Unit Inventory for Sacramento County at: http://www.sacog.org/infoctr/pophsg/rads/sac/sacr.html. The housing units of 1980 – 89 is calculated from the annual C40 residential housing units authorized by building permits in the following manner:

S_t = S_t-1 + N_t – DS_t-1

Where

S_t represents stock of housing in period t

S_t-1 represents stock of housing in period t – 1

D represents the depreciation rate

N_t represents the number of housing permits in period t

S_t are known for the period 1990 - 2000, and N_t are known for the period 1980 – 2000. It is easy to calculate D as equal to .008, which is the average depreciation rate for the period 1990 – 2000. Using .008 as a proxy for the depreciation rate for the period of 1980 – 1989,

S_t-1 = S_t – N_t + .008 S_t-1 …….(1)

S_t-1 = (S_t – N_t)/.992 …….(2)

Using equation two, it is easy to reconstruct the housing stock for the period 1980–1989.

The annual new housing units authorized and the valuation of these units in Sacramento County is taken from the U.S. Census Bureau website, http://www.census.gov/const/www/c40/table3.html. Population numbers for the period 1980 – 1989 are from: www.census.gov/population/www/estimates/countrypop.html. The population for the period 1990 – 2000 is taken from 2000 Population U.S. Estimates and the Housing Unit Inventory of Sacramento County from http://www.sacog.org/infoctr.pophsg/cities/sac/sacr.htm. The median price of home resale in Sacramento County is taken for the period 1983 – 1997 from http://www.sacbee.com/ourtown/economy/housing.html. Per capita disposable income of California is used as a proxy for Sacramento County for the period 1980 – 2000. The source of per capita disposable income is from Table D-7, U.S Department of Commerce, Bureau of Economic Analysis website http://www.beadoc.gov/. Employment cost index in the west for the period 1980 – 2000 is taken from employment cost index, Bureau of Labor Statistics Report as a proxy for construction costs, http://stats.bls.gov/ecthome.htm. This is an acceptable method since labor costs represent a stable proportion of residential housing costs.

The mortgage rates are taken from historical mortgage rates for the period 1983 – 2000, HSH Associates’ website http://hsh.com/mtghst.html. The site provided national monthly mortgage rates from which the annual mortgage rate was calculated.

This model is formalized in a fairly straightforward manner. The model is divided into two structural equations of demand and supply functions and the reduced form equation for investment in residential housing. There are explicit and implicit assumptions that have been made through the construction of the demand and supply equations. The explicit assumptions are in the linear functional form, in the variables included, and in the expected sign of the coefficients. The implicit assumptions are that human economic behavior can be explained in mathematical form, and all variables not included in the equation are held constant. The constant term implies that the mean effect of the omitted variables is captured by the constant term, and the variability of the omitted variables is captured by the variance of the error term. The ordinary least square estimate technique is used to estimate the investment function of residential housing.

Misspecification: It actually is impossible to be sure that our model or  any other  model  is correctly specified, but it is important to approximate the actual process using all the available means such as theoretical, special events (war-peace), statistical, and intuitive analysis to build the model. Caudill (1988) argued that the functional form, the use of proxy variables, the inclusion of social and cultural variables, and the choice of measures for a variable (i.e., the use of adjustable rate mortgage or fixed rate mortgage for mortgage rate) are not based on theoretical grounds, but statistical factors.  There are many types of misspecification such as excluded relevant variables, included irrelevant variables, functional forms, and misspecification association with error terms.

Theoretical Evaluation of Including a New Variable in the Model: Looking  at the effect of a new variable on the  regression can help to decide whether to include or exclude the new variable. The effect of a new variable on the regression can produce one of the four following results:

First: The coefficient of a new variable is insignificant and has little effect on the other coefficients. If this is the case, then the new variable should not be included unless the theory requires it.

Second: The coefficient of the new variable is insignificant but has a substantial effect on other coefficients by making some of them insignificant.  This multicollinearity problem should be corrected before one includes or excludes the new variable in the regression.

Third: The new variable has a significant coefficient and has a strong influence on the other coefficients. This influence suggests that one should include the new variable after correcting for the multicollinearity problem.

Fourth: The optimal situation of including the new  variable arises  when the coefficient of the new variable  is significant and other variables’ coefficients did not change substantially.

Lagged population and lagged labor cost index variables were introduced to the model and found to be statistically significant. The inclusion of these two variables improved the forecast accuracy of the model. However, the inclusion of two additional independent variables increased the number of the independent variables to seven variables. This makes the model vulnerable to the multicollinearity problem.

Multicollinearity:  This kind of problem arises when one has independent variables in a regression that are highly correlated. This is a common problem when one has more than five independent variables.  The problem with multicollinearity is that the collinear data will not add additional information not already known. It is also difficult to know the separate influence of each explanatory variable.   Also, collinearity arises not only between the independent variables, but also between one independent variable and the constant term.  Multicollinearity will produce biases in the individual parameters but not in the sum of the parameters. Different degrees of multicollinearity exist with all multivariate regression analysis. Thus, resolving the problem of multicollinearity is necessary only if the independent  variables are highly correlated. A rule of thumb mentioned by Farrar and Glauber (1967) states that if the correlation between two independent variables exceeds .8 or .9, then one should have multicollinearity problems. Multicollinearity will affect the stability of the coefficients so that different parts of the data will give substantially different results. It also leads to an incorrect theoretical sign, a high F-value, and low t-values for some or all of the coefficients. Multicollinearity makes regression  results sensitive to the deletion of one row of observations or a  column of  the  independent  variables. Sometimes it also leads to wrong magnitudes of a coefficient or making the magnitude meaningless (such as when marginal propensity to consume is more than one). Multicollinearity also leads to a very high R² value while some or all of the coefficients are insignificant. When forecasting, the existence of multicollinearity in the model will prevent one from producing a successful forecast, unless a stable dependency relationship between independent variables continues during future observations. In other words, multicollinearity will have no effect on the forecasting model, if and only if, the multicollinearity continues into the period predicted.

Correction of multicollinearity: One way  to deal with this problem is by making a linear combination of the collinear variables if it has a meaningful interpretation and by using that combination as a regressor. The combination of the mortgage rate and the change of the median price of home resale to produce capital gain is an appropriate application of this principle. The investment models were checked for multicollinearity to see if the correlation between two independent variables exceeds .8 or .9 and none were found.

In the basic model there is a demand and supply function and a reduced form equation for investment. The mathematical forms of the model are as follows:

Demand Equation: D_t = f(DI _t+1, %change ARM _t+1, % change P_t)

Where

DI _t+1 represents per capita disposable income in period t +1;

ARM _t+1 represents adjustable mortgage rate in period t +1;

P_t represents real median price of home resale in Sacramento County.

Supply Equation: S_t = f(% change P_t , % change ARM _t+1, % change C_t-1, S_t-1, I_t)

Where

C_t-1 represents employment cost index in period t -1;

S_t-1 represents housing units in period t -1;

I_t represents investment in residential housing in period t.

The Reduced Form Equation: I_t = f(DI _t+1, % change ARM _t+1, % change P_t, % change C_t-1, S_t-1)

The Estimated Equation:

I = 2079737.1+ 50.74 DI _t+1- 48948.09 % change ARM _t+1 + 5041673 % change P_t

t-value 4.051 .996 -.144 7.704

-2.65 -1.033

DW = 2.325 F = 18.049 Adjusted R Square = .859

The result of the regression in model one shows an insignificant coefficient of interest rate and disposable income. Therefore, capital gains (CG_t) was used instead of percentage change in ARM and percentage change in P, where CG_t = % change P_t - ARM_t.

The new model is a modified model from the basic model in which capital gains replaced percentage change in P_t and mortgage rate.

Demand Equation: D_t = f(DI _t+1, CG_t)

Supply Equation: S_t = f(CG_t, % change C_t-1, S_t-1, I_t)

Reduced Form Equation: I_t = f(DI _t+1, % change C_t-1, S_t-1, CG_t )

The Estimated Equation:

I_t = 2566083.3 + 80.665 DI _t+1 - 13000000 % change C_t-1 - 5.62 S_t-1 + 50104.542 CG_t

t-value 5.592 1.786 -2.658 -2.034 8.362

DW = 2.357 F = 26.183 Adjusted R Square = .878

The third model is developed to improve the forecast accuracy by introducing two additional variables to the basic model (model 1). The additional variables are lagged population and lagged employment cost index.

Demand Equation: D_t = f(DI _t+1, FRM_t, change P_t , POP_t-1)

Where

POP_t-1 represents population of Sacramento County in period t-1.

Supply Equation: S_t = f(change P_t , FRM_t , % change C_t-1, change C_t-1, S_t-1, I)

Reduced Form Equation: I_t = f(DI _t+1, FRM_t, change P_t, POP_t-1,% change C_t-1, change C_t-1, S_t-1)

The Estimated Equation:

I = 2516418.9 + 484.327 DI _t+1 - 161992.7 FRM_t + 37.659 change P_t +14.578 POP_t-1

t-value 2.149 4.485 -2.525 5.716 2.516

DW = 2.476 F = 29.555 Adjusted R square = .935

The absolute average forecast inaccuracy (AAFI) as well as the absolute percentage forecast inaccuracy (APFI) over 15 years as an instrument to rank the accuracy of each model follows:

Model AAFI (000) APFI

1. 90321 4.8%

2. 78947 4.19%

3. 55041 2.92%

The sign of all the coefficients were as expected. The impact of expected disposable income, population, capital gain, and change in real median prices of home resale is positive. While the effect of mortgage rates, stock of housing, and employment cost index is negative, as anticipated. Based on the relatively simple models and the weak data base, the findings from these models were encouraging. The estimated reduced form equations represent satisfactory statistical fits of the t-value, f-value, adjusted R square, and expected sign of the coefficients of regressions. Durban-Watson (DW), as a measure of correlation of the error term, shows that no correlation exists in any of these estimated equations. If there is an interest in measuring the influences of percentage change of real median housing prices and the percentage change in mortgage rate, then the individual would use Model One (Basic Model). But if there is an interest in the net impact of capital gains on housing investment, then Model Two (Modified Model) should be used. Finally, if the interest is purely forecasting, then Model Three (Extension of Basic Model) will be relevant.

Modeling the residential housing investment for Sacramento County is challenging due to the lack of the appropriate data and the use of proxy variables. But theses models were successful for forecasting residential investment based on AAFI, APFI, expected sign of coefficients, and statistical fits. The models provided useful information on the dynamic of residential housing in Sacramento County.

Baffoe-Bonnie, John. (1998) "The Dynamic Impact of Macroeconomic Aggregates on Housing Prices and Stock of Houses: A National and Regional Analysis." Journal of Real Estate Finance and Economics, 17:2, 179-197.

Bureau of Labor Statistics Report (2001) "Employment Cost Index." [On-line]. Available: http://stats.bls.gov/ecthome.htm.

Chan, Thomas Sai-Fan. (1999) "Residential Construction and Credit Market Imperfection." Journal of Real Estate Finance and Economics, 18:1, 125-139.

Caudill, S. B. (1988) "The Necessity of Mining Data." Atlantic Economic Journal, XVI(3), 11-18.

Dipasquale, Denise (1999) "Why Don't We Know More About Housing Supply?" Journal of Real Estate Finance and Economics, 18:1, 9-23.

Dipasquale, Denise, and William C. Wheaton (1994) "Housing Market Dynamics and the Future of Housing Prices." Journal of Urban Economics, 35(1), 1-27.

Farrar, D. E., and Robert R. Glauber (1967) "Multicollinearity in Regression Analysis: The problem Revisited." Review of Economics and Statistics, 49(1), 92-107.

HSH Associates (2000) "Historical Mortgage Rates for the Period 1983 – 2000." [On-line]. Available: http://hsh.com/mtghst.html.

Hu, Joseph (1992) "Housing and Mortgage Securities Markets: Review, Outlook, and Policy Recommendations." Journal of Real Estate Finance and Economics, 5: 167-179.

Lereah, David. (1997) "Housing Finance: A Long Term Perspective." Business Economics, July, Vol. XXXII, Number 3, 21-26.

Mankiw, N.G, and Weil, D.N. (1989) "The Baby Boom, The Baby Bust, and The Housing Market." Regional Science and Urban Economics, 19(2), 235-258.

Phillips, Richard A. (1999) "Two-Earner Households and Housing Demand: The Effect of the Wife's Occupational Choice." Journal of Real Estate Finance and Economics, 4: 83-91.

Plaut, Steven E. (1991) "The 'Rent-To-Own' Mortgage and Prepayment Risk." Journal of Real Estate Finance and Economics, 4: 69-81.

Potepan, Michael (1996) "Explaining Intermetropolitan Variation in Housing prices, Rents, and Land Prices." Real Estate Economics, 24 (Summer), 219-246.

Poterba, James M. (1984) "Tax Subsidies to Owner Occupied Housing: An Asset Market Approach." Quarterly Journal of Economics, 99(4), 729-52.

Sacramento Bee (2001) "The median price of home resale in Sacramento County." [On-line]. Available: http://www.sacbee.com/ourtown/economy/housing.html.

Sacramento County (2000) Population and Housing Unit Inventory. [On-line]. Available: http://www.sacog.org/infoctr/pophsg/rads/sac/sacr.html.

Sacramento County (2000) "2000 Population U.S. Estimates and the Housing Unit Inventory of Sacramento County." [On-line]. Available: http://www.sacog.org/infoctr.pophsg/cities/sac/sacr.htm.

Sieders, David F. (1997) "Trends and Cycles in Housing Production." Business Economics, July, Vol. XXXII, Number 3, 12-16.

Somerville, C. Tsuriel. (1999) "Residential Construction Costs and the Supply of New Housing: Endogeneity and Bias in Construction Cost Indexes." Journal of Real Estate Finance and Economics, 18:1, 43-62.

Topel, Robert, and Sherwin Rosen (1988) "Housing Investment in the united States," Journal of Political Economy, 96(4), 718-740.

U.S. Census Bureau (2000) "The Annual New Housing Units Authorized and The Valuation of These Units." [On-line]. Available: http://www.census.gov/const/www/c40/table3.html.

U.S. Census Bureau (2000) "Population Estimates." [On-line]. Available: www.census.gov/population/www/estimates/countrypop.html.

U.S. Department of Commerce (2000) "Per Capita Disposable Income." [On-line]. Available: http://www.beadoc.gov/. Table D-7.