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Path Analysis

Human Capital vs Homeownership

Gaetan Guy Lion

April 2009


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Introduction

Path Analysis is a way to decompose

correlations between different variables, in

this case Human Capital vs Homeownership

rate.

Human Capital is % of population over 25with a college degree.


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The Rosetta Stone in Path Analysis

With standardized variables within a single relationship the

Correlation is equal to the Slope.

Correlation = COVAR (X, Y)/(sX)(sY)

Slope = COVAR (X, Y)/VAR X

If both s= 1. Correlation = Slope


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The Path Analysis Diagram

The Path Analysis Diagram defines our hypothesis. Human Capital has an impact on:

Home Affordability (-) as highly educated wage earners bid up prices of homes,

Demographic/youth (-) more youth fewer older people with degrees, and

Unemployment (-) as Human Capital lowers unemployment.

In turn, those intermediary variables impact Homeownership rate:

Housing Affordability (+), if homes are more affordable homeownership goes up.

Demographic-Youth (% of population between 20 and 29), (-) as younger people can

ill afford homes, andUnemployment (-) as unemployed lack the income to buy homes.

Independent Intermediary Dependent

variable variables variable

Housing

- affordability +

Human - Demographic - HomeCapital (youth) ownership

- -

Unemployment


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The Actual Correlations

Correlations -0.176

Housing

-0.181 affordability 0.573

Human -0.064 Youth -0.249 Home

Capital ownership-0.250 Unemploy. 0.064

We embedded the correlations within the diagram. We also added a correlation

directly from Human Capital to Home ownership. Most correlation signs supportthe hypothesis except Unemployment.


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The Path Coefficients

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.633

R Square 0.401

Adjusted R Square 0.375

Standard Error 0.785

Observations 111

Coefficients Stand. Error t Stat P-value

Human capital -0.130 0.078 -1.665 0.099

Housing affordability 0.581 0.079 7.334 0.000

Young population -0.228 0.077 -2.976 0.004

Unemployment -0.181 0.081 -2.227 0.028

Given that the variables are standardized, all bivariate correlations already

represent Path coefficients (in white). Well calculate the Path coefficients in

yellow with a regression model.

Dependent variable is Homeownership rate

Path Coefficients

-0.130

Housing



Capital ownership

-0.250 Unemploy. -0.181


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Correlations vs Path Coefficients

Correlations -0.176

Housing



Capital ownership

-0.250 Unemploy. 0.064

Correlations reflect the relationship between just two variables. The Path

coefficients reflect the effect one variable has on another when controlled

for the other three variables. Now the Path coefficient of Unemployment

rate is negative.

Path Coefficients

-0.130

Housing



Capital ownership

-0.250 Unemploy. -0.181


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Direct and Indirect Effects

Correlation

Direct Effect Indirect Effect

Causal Effect

The Correlation of the independent variable can be decomposed into its

Direct Effect and Indirect Effect on the dependent variable. The Causal

Effect is the sum of the mentioned Effects and should equal the

Correlation.


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Human Capital

Direct and Indirect Effects

Human Capital causal effect

(-0.176) on Homeownershipequals its correlation.

Path Coefficients

-0.130Housing



Capital ownership

-0.250 Unemploy. -0.181

Decomposing correlations into indirect and direct effects

Human Capital indirect effect on Homeownership A B A x B

Human Cap. -> Housing affordability -> Homeownership -0.181 0.581 -0.105

Human Cap. -> Youth -> Homeownership -0.064 -0.228 0.015

Human Cap. -> Unemployment -> Homeownership -0.250 -0.181 0.045

-0.045

Human Capital direct effect on Homeownership -0.130

Human Capital Causal effect on Homeownership

a) Indirect effect -0.045

b) Direct effect -0.130

Total causal effect -0.176


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Assessing the Model Fit

Assessing the model fit entails:

1. Reconstructing all the correlations by using the

path coefficients; and

2. Assessing the closeness of the fit between thereconstructed correlations and actual ones.


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Reconstructing correlations

Reconstructing the other correlations vs Homeownership

Housing affordability A B C A x B x C

-- Homeownership 0.581 0.581-- Human capital -- Homeownership -0.181 -0.130 0.024

-- Human capital -- Youth -- Homeownership -0.181 -0.064 -0.228 -0.003

-- Human capital -- Unemploy.-- Homeownership -0.181 -0.250 -0.181 -0.008

0.593

Youth

-- Homeownership -0.228 -0.228

-- Human capital -- Homeownership -0.064 -0.130 0.008-- Human capital -- Housing afford. -- Homeownership -0.064 -0.181 0.581 0.007

-- Human capital -- Unemployment -- Homeownership -0.064 -0.250 -0.181 -0.003

-0.216

Unemployment

-- Homeownership -0.181 -0.181

-- Human capital -- Homeownership -0.250 -0.130 0.033

-- Human capital -- Housing afford. -- Homeownership -0.250 -0.181 0.581 0.026

-- Human capital -- Youth -- Homeownership -0.250 -0.064 -0.228 -0.004

-0.126

Path Coefficients

-0.130

Housing



Capital ownership

-0.250 Unemploy. -0.181


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Assessing closeness of fit with RMSE

Root Mean Square Error (RMSE)

Standard Error of Prediction

Correlation vs Homeownership Actual R Predicted R Error Error 2

Human capital -0.176 -0.176 0.000 0.0000

Housing affordability 0.573 0.593 -0.020 0.0004

Youth -0.249 -0.216 -0.033 0.0011

Unemployment rate 0.064 -0.126 0.190 0.0361

RMSE 0.0969


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Path Analysis next step

We test other models and check if their fit is

superior to the original model.

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