City Story: House Prices and Competitiveness

Abstract

In this case study, we consider technology jobs and housing affordability trends in the Silicon Valley region. We estimate a vector error correction model using measures of housing affordability, GMP, start-ups and patent applications. Despite ever-higher housing prices and declining affordability, innovation, as measured by the number of technology jobs, as well as other metrics, continues to thrive.

Appendices

Appendix on Econometrics Analysis

1.1 Linear Regression Analysis

We conduct regression analysis to see how urban competitiveness affects housing price. We include various measures of competitiveness. Instead of using the original series, we first detrend the data by transforming them into the growth rates. We apply the linear regression model to each MSA separately.

Data are collected from multiple sources. HPI is the seasonally adjusted housing price index from FHFA, deflated by consumer price index of all urban consumers (housing in San Francisco-Oakland-San Jose, CA). Data on population is from US Census Bureau. Productivity is measured by annual percent change in real gross product per worker from Bureau of Labor Statistics, Bureau of Economic Analysis and Moody’s Analytics. Employment data comes from Bureau of Labor Statistics. Start-up growth relies on the data from Kauffman Foundation and Business Dynamics Statistics.

Two MSAs exhibit different pattern of the housing price and competitiveness. We didn’t find significant correlation between GMP growth and HPI growth, nor is the relation between productivity growth and housing price growth statistically significant. In both MSAs, population growth does exert a negative impact on the housing price growth. The response to employment growth and start-up growth are different across two MSAs. In San Francisco, we find the effect of both factors positive and significant on housing price growth, while in San Jose, there is no convincing evidence in favor of the claim. Although our time series are short, more than 50% of the variation in housing price growth can be explained in our model (Table 7.31).

Table 7.31 Linear model

Availability of data at MSA level does restrict our regression results. To augment our analysis, we look at the correlation matrices. In the matrix, we include HPI growth, GMP growth, personal income growth, population growth, patent growth, productivity growth, start-up growth, employment growth, growth of employment in technology sector and unemployment rate. We didn’t find strong evidence that higher patent growth will be associated with housing price growth. The correlation between HPI growth and productivity growth in San Jose is positive and significant with 95% confidence, while such correlation in San Francisco is only significant at 90% confidence. Start-up growth is not so correlated with housing price growth in San Jose, but the effect is significant in San Francisco. Though employment share of technology sector is extremely high in those two MSAs, we don’t find strong evidence that higher employment growth in technology sector is associated with higher housing price growth in two MSAs (Tables 7.32 and 7.33).

Table 7.32 Correlation matrix, San Jose

Table 7.33 Correlation matrix, San Francisco

1.2 Vector Autoregression Analysis

• Variable Choice

To explore the relation between urban competitiveness and local housing prices, we look at the interaction and dynamics of the following three variables in the benchmark analysis: the seasonally adjusted housing price index from FHFA deflated by consumer price index of housing, the start-up density, and real gross metropolitan product. We use the log variables, instead of the originals, to map those variables from the positive domains to a comparable range defined on the real line.

Start-up density is a proxy of innovative potential and vibrancy of local industries, driving the future growth of a city. The indicator reflects the state of both the demand and the supply side of the local labor market, in terms of business and job creation activities. Housing price index summarizes the sales and refinancing of the local housing market, which is the key statistic of the report. Real gross metropolitan product is defined as the market value of final goods and services created for a given period, which indicates economic performance as well as labor productivity of the local market.

We focus our attention on those three variables for two reasons. First, as the frequency of the time series is annual and the length of the data is short due to data availability, we try to work on a simple model with fewer but necessary variables to understand the relationship between competitiveness and housing prices. Second, those three time series seem to describe the economic dynamics well. Our post-estimation tests and robustness check show that the qualitative features over time as well as across MSAs can be captured by the simple dynamics of the benchmark model.

• Model Choice

We consider applying vector autoregression (VAR) or vector error correction (VECM) model to our analysis. Instead of taking a stand on the dependency or linkage between competitiveness and housing prices, we attempt to treat them equally by including all contemporaneous variables as dependent and their lags as explanatory variables. We estimate the system as a whole.

Before going to regression analysis, we conduct Dicky-Fuller test on each series to test whether the processes are unit-root, and confirm that they are not covariance stationary, but integrated series of order 1. Hence, it is improper to apply the VAR model directly to the level variables. We further test whether the series are cointegrated, Johansen’s trace statistics show that we cannot reject the existence of 1 cointegrating relationship between our time series. It is improper to apply the first differences of the series to a VAR model which assumes no cointegrating relationship. A VECM model will correct the problem by introducing an error correction term in each first-difference VAR regression equation.

• Model Specification

We consider the following Vector Error Correction model (VECM) that describes the evolution of k variables over the sample period from 1991 to 2014 on the annual basis

$$ \Delta {\mathbf{y}}_{\text{t}}^{i} = {\varvec{\Pi}}^{i} \varvec{y}_{t}^{i} + \sum \limits_{j = 1}^{J - 1} {\varvec{\Gamma}}_{j}^{i} \Delta {\mathbf{y}}_{t - j}^{i} + {\varvec{\upepsilon}}_{\text{t}}^{i} $$

where ${\mathbf{y}}_{\text{t}}^{i} = \left( {{\text{y}}_{{1{\text{t}}}}^{i} ,{\text{y}}_{{2{\text{t}}}}^{i} , \ldots ,{\text{y}}_{\text{kt}}^{i} } \right)^{{\prime }}$ and $\varvec{\upepsilon}_{\text{t}}^{i} = \left( {\epsilon_{1t}^{i} , \ldots ,\epsilon_{kt}^{i} } \right)^{{\prime }}$ are column vectors of length k = 3, and ${\varvec{\Gamma}}_{j }^{i}$ is a ${\text{k}} \times {\text{k}}$ matrix. In our setting,

$$ {\mathbf{y}}_{\text{t}}^{i} = \left( {\log \left( {HPI_{t}^{i} } \right),\log \left( {GMP_{t}^{i} } \right),\log \left( {Startup_{t}^{i} } \right)} \right)^{{\prime }} $$

Superscript $i$ stands for San Jose or San Francisco Metropolitan Statistical Area, while subscript indicates the lag of the year. ${\varvec{\Pi}}^{i} \equiv\varvec{\alpha}^{i}\varvec{\beta}^{{i{\prime }}}$ is the ${\text{k}} \times {\text{k}}$ error correction matrix, made up of a ${\text{k}} \times {\text{r}}$ matrix $\varvec{\alpha}^{i}$ and an ${\text{r}} \times {\text{k}}$ matrix $\varvec{\beta}^{{i{\prime }}}$. Both $\varvec{\alpha}^{i}$ and $\varvec{\beta}^{{i{\prime }}}$ have full rank r = 1. By Granger Representation Theorem, the cointegrating relationship is a linear combination of all variables

$$ {\text{z}}^{i} =\varvec{\beta}^{{i{\prime }}} \varvec{y}^{i} $$

${\text{J}} = 2$ the order of the VECM model, or the maximum lags to be included. It is chosen based on minimizing Schwarz Bayesian information criterion (SBIC).

• Estimation Results

The estimation results of VECM for each MSA are summarized in the table. The first term in each regression, L.ce1, denotes the error correction, or $\alpha^{i}$ In terms of notation above. We observe a significant coefficient of error correction term in the regression of the start-up density, confirming our finding that the time series are cointegrated. All the coefficients in $\Gamma ^{i}$ are smaller than unity. The stability of the system is thus guaranteed.

By comparing the estimated coefficients between San Jose and San Francisco, we notice most of the coefficients carry the same sign or are qualitatively the same. But two MSAs do differ in several aspects. Besides, we observe the effect of last year’s GMP growth on today’s housing price growth is negative in both San Jose and San Francisco. The reaction to GMP is not statistically significant, so it may stem from the length of the data we work with.

Another noticeable difference is the effect of last year’s start-up density growth on today’s GMP growth. The coefficient in San Jose is negative and that in San Francisco is positive, though both are not statistically significant.

We also observe significantly positive effect of housing price growth and GMP growth on the start-up density growth. 1% increase in the housing price growth will boost start-up density growth by 0.37% in San Jose and 0.38% in San Francisco, almost identical across two MSAs. 1% increase in GMP growth will boost start-up density growth by 0.49% in San Jose and 0.37% in San Francisco (the latter is not significant).

The coefficients in $\Gamma$ capture the short-run dependence of the lag variables. As to the long run, the cointegration equation is informative. Our model implies the following cointegrating relationship in the long run (Table 7.34):

Table 7.34 VECM model

$$ \begin{aligned} {\text{z}}^{\text{SJ}} & \equiv \log \left( {HPI^{SJ} } \right) - 2.311\log \left( {GMP^{SJ} } \right) - 5.105\log \left( {Startup^{SJ} } \right)\sim I\left( 0 \right) \\ {\text{z}}^{\text{SF}} & \equiv \log \left( {HPI^{SF} } \right) - 2.839\log \left( {GMP^{SF} } \right) - 3.353\log \left( {Startup^{SF} } \right)\sim I\left( 0 \right) \\ \end{aligned} $$

where both ${\text{z}}^{\text{SJ}}$ and ${\text{z}}^{\text{SF}}$ are covariance stationary series, or ${\text{I}}(0)$. In both MSA, GMP and the start-up density are positively correlated with local housing prices in the long run. In San Jose, 1% increase in GMP (or start-up density) is associated with 2.3% increase (or 5.1%) in the housing price, ceteris paribus. In San Francisco, 1% increase in GMP (start-up density) is associated with 2.8% increase (3.4%) in the housing price, ceteris paribus. All the coefficients are highly statistically significant at 99% confidence.

If both GMP and start-up density increase by 1%, we will witness roughly the same percentage increase (6–8%) in the housing price across MSAs. Nevertheless, the elasticity of the housing price with respect to GMP and start-up density are different across MSAs. The housing price in San Jose is more responsive to the change of start-up density, while the housing price in San Francisco responds more to GMP (Table 7.35).

Table 7.35 Cointegration equation

• Impulse Response Functions (IRF)

To explore the dynamics of the system, we simulate the model by hitting each variable with a one-time one-standard-deviation shock to see how each variable will react to the unexpected impulse over time. The figures show the panels of impulse response functions for each MSA. The length of each step is by year. The vertical axis shows the change of the response variable. Due to the cointegrating relationship, the effect of a shock in most cases are permanent, though it will gradually settle down to the new level in 3–5 years.

Panel (row 1, col 3) tracks the response of the start-up density to an impulse of GMP. We can see the response variable decreases. The initial increase is driven by the short-run positive relationship shown in the start-up regression equation. But the long-run negative response is mainly driven by the cointegrating relation.

Panel (row 3, col 1) plots the response of GMP to a start-up density shock. IRF implies a negative response, but the estimated coefficient of the start-up density in GMP regression is insignificant for both MSAs, leading to a wide confidence interval. We cannot conclusively say that the response is negative and significant.

Panel (row 2, col 1) and Panel (row 2, col 3) show the response to the unexpected shock of the housing price. Higher housing prices boosts both GMP and start-up density. The effect is more pronounced in San Jose than in San Francisco.

Panel (row 3, col 2) show that a one-standard deviation shock to the start-up density results in 2% increase in housing price in both MSAs.

Panel (row 1, col 2) shows the reaction of the housing price to a GMP shock. The effect is predicted to be negative in San Jose, while the effect in San Francisco, is initially positive. As the estimated coefficient of GMP in HPI regression is insignificant, the confidence interval will be fat, so that the difference in IRF across MSAs is inconclusive of the different reaction to a GMP shock across MSAs (Figs. 7.77 and 7.78).

Fig. 7.77

Impulse response function—San Jose

Fig. 7.78

Impulse response function—San Francisco

Appendix: The Calculation of Two Indexes Measuring the Housing Affordability in Tokyo

Year	Income	Size	Pincome	Price	Fspace	Price30	Index 1	Index 2
	Average household income (10,000 Yen)	Average household size (person)	Per capita income (10,000 Yen)	Average dwelling price (10,000 Yen)	Average floor space (m2)	Price for 30 m3 (10,000 Yen)	Price30 to Pincome	Price to income
1975	327	3.1	104.1	1530	56.8	808.1	7.8	4.7
1976	361	3.1	115.3	1630	56.6	864.0	7.5	4.5
1977	400	3.1	128.2	1646	56.4	875.5	6.8	4.1
1978	412	3.1	132.5	1711	56.1	915.0	6.9	4.2
1979	445	3.1	143.1	1992	59.5	1004.4	7.0	4.5
1980	493	3.1	159.0	2477	63.1	1177.7	7.4	5.0
1981	516	3.1	167.5	2616	61.0	1286.6	7.7	5.1
1982	534	3.1	174.5	2578	60.2	1284.7	7.4	4.8
1983	557	3.1	182.6	2557	59.8	1282.8	7.0	4.6
1984	594	3.0	195.4	2562	61.1	1257.9	6.4	4.3
1985	634	3.0	209.2	2683	62.8	1281.7	6.1	4.2
1986	663	3.0	221.0	2758	65.0	1272.9	5.8	4.2
1987	660	3.0	222.2	3579	65.2	1646.8	7.4	5.4
1988	682	2.9	232.0	4753	68.0	2096.9	9.0	7.0
1989	730	2.9	250.0	5411	67.9	2390.7	9.6	7.4
1990	767	2.9	264.5	6123	65.6	2800.2	10.6	8.0
1991	828	2.9	287.5	5900	64.9	2727.3	9.5	7.1
1992	875	2.8	308.1	5066	63.3	2400.9	7.8	5.8
1993	854	2.8	303.9	4488	63.8	2110.3	6.9	5.3
1994	854	2.8	309.4	4409	64.6	2047.5	6.6	5.2
1995	856	2.7	315.9	4148	66.7	1865.7	5.9	4.8
1996	842	2.7	314.2	4238	69.5	1829.4	5.8	5.0
1997	853	2.7	321.9	4374	70.3	1866.6	5.8	5.1
1998	896	2.6	342.0	4168	71.0	1761.1	5.1	4.7
1999	859	2.6	330.4	4138	71.8	1728.0	5.2	4.8
2000	815	2.6	317.1	4034	74.7	1620.0	5.1	4.9
2001	813	2.6	318.8	4026	77.0	1569.0	4.9	5.0
2002	823	2.5	326.6	4003	78.0	1539.0	4.7	4.9
2003	783	2.5	313.2	4069	74.7	1634.1	5.2	5.2
2004	796	2.5	321.0	4104	74.6	1650.0	5.1	5.2
2005	790	2.5	321.1	4107	75.4	1635.0	5.1	5.2
2006	794	2.4	325.4	4200	75.7	1664.5	5.1	5.3
2007	798	2.4	331.1	4644	75.6	1842.9	5.6	5.8
2008	791	2.4	332.4	4775	73.5	1949.0	5.9	6.0
2009	804	2.3	343.6	4535	70.6	1927.1	5.6	5.6
2010	762	2.3	331.3	4716	71.0	1992.7	6.0	6.2
2011	742	2.3	325.4	4578	70.5	1949.2	6.0	6.2
2012	759	2.3	335.8	4540	70.4	1933.8	5.8	6.0
2013	782	2.2	349.1	4929	70.8	2089.4	6.0	6.3
2014	775	2.2	349.1	5060	71.2	2133.2	6.1	6.5
2015	786	2.2	357.3	5518	70.8	2337.8	6.5	7.0

1. Source Calculated by the author based on the statistical data of MILT (2016)