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Consumer Spending causing Unemployment?...

Guy Lion

January 2010

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Introduction

The causal hypothesis is whether change in Consumer Spending causes Unemployment [rate] or vice versa…

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Demonstrating Causality

• Is the causal association strong?

• Does it make sense in an economics context?

• Is it temporally correct? Does the cause precede the effect?

• Can the causal effect be replicated using different time series?

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Statistical methods to demonstrate causality

• Granger causality consists in running a series of linear regressions and conducting hypothesis tests to measure the difference in residuals (more details on next slides);

• Path analysis is another method most appropriate when an independent variable has an impact on intermediary variables that in turn impact the dependent variable.

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Granger Causality

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Granger Causality steps

1. Develop a Base case autoregressive model using dependent variable and its lagged values as the independent variable.

2. Develop a Test case model by adding a second lagged independent variable you want to test.

3. Calculate the square of the residual errors for the two models and run an unpaired student t Test to check if the residuals are significantly lower when you add the tested second variable.

4. Redo steps 1 through 3, but reverse the direction. By comparing the P value, you can see if A Granger causes B more than B Granger causes A.

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Granger Causality – The Basics

A Granger causes B

Base case Model Test ModelAutoregressive Multivariate

X1 = Lag B X1 = Lag BY = B X2 = Lag A

Y = B

Square Residuals Square Residuals

Hypothesis testing

F or t TestDo the 2 samples ofresiduals come from same population?

Linear Regression

Here, the change in Consumer Spending would be variable A. And, the Unemployment rate would be variable B.

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Granger Causality – the whole picture

A Granger causes B B Granger causes A

Base case Model Test Model Base case Model Test ModelAutoregressive Multivariate Autoregressive Multivariate

X1 = Lag B X1 = Lag B X1 = Lag A X1 = Lag AY = B X2 = Lag A Y = A X2 = Lag B

Y = B Y = A

Square Residuals Square Residuals Square Residuals Square Residuals

Hypothesis testing Hypothesis testing

F or t Test F or t TestDo the 2 samples of Do the 2 samples ofresiduals come from residuals come from same population? same population?

Granger CausalityCompare significance of F or t Stat P value to find out if A causes B more than B causes A.

Linear Regression Linear Regression

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Correlations with quarterly lags

Since 1970. Quarterly dataCausal Unempl. Consumer sp.

Dependent Consumer sp. Unempl.

Lag 1 0.19 -0.08Lag 2 0.24 -0.18Lag 3 0.25 -0.27Lag 4 0.27 -0.33Lag 5 0.26 -0.34Lag 6 0.24 -0.37Lag 7 0.23 -0.36Lag 8 0.20 -0.32

Since 1990. Quarterly dataCausal Unempl. Consumer sp.

Dependent Consumer sp. Unempl.

Lag 1 -0.16 -0.44Lag 2 -0.05 -0.53Lag 3 0.05 -0.62Lag 4 0.10 -0.61Lag 5 0.18 -0.58Lag 6 0.21 -0.56Lag 7 0.21 -0.49Lag 8 0.20 -0.44

The correlations tables with quarterly lags from 1 quarter to 8 quarters indicate where the causality between the two variables appears more pronounced. In terms of direction (sign), we get far more logical results when we focus on quarterly change in Consumer Spending causing Unemployment [rate] on the right rather than the reverse on the left. When we cut the data since 1990 instead of since 1970, we get stronger correlations. As shown, the more meaningful correlations are in the highlighted yellow zone. And, the strongest (negative) correlation is for change in Consumer Spending impacting the Unemployment rate level with a three quarter lag. So, that’s the relationship we will study.

Note the variables are seasonally adjusted. Thus, the lag 4 quarter has no competitive advantage over any other lags.

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Granger Causality test (data since 1990)

This is a very successful Granger Causality outcome as it demonstrates the direction of the Granger causality is clearly Consumer Spending -> Unemployment and not the reverse. The P value (unpaired t test) of only 11.0% confirmed the (squared) residuals were materially lower for the Test model vs the Base case model. When the causality direction was reversed (Unemployment -> Consumer Spending), the P value jumped to 89.8% suggesting that any difference in (squared) residuals between the Base case Model and Test Model was pretty much due to randomness.

A Granger causes B B Granger causes A

Base case Model Test Model Base case Model Test ModelAutoregressive Multivariate Autoregressive Multivariate

X1 = Lag B X1 = Lag B X1 = Lag A X1 = Lag AY = B X2 = Lag A Y = A X2 = Lag B

Y = B Y = A

Square Residuals Square Residuals Square Residuals Square Residuals

Hypothesis testing Hypothesis testing

F or t Test F or t TestDo the 2 samples of Do the 2 samples ofresiduals come from residuals come from same population? same population?

Granger CausalityCompare significance of F or t Stat P value to find out if A causes B more than B causes A.

Linear Regression Linear Regression

Consumer Spending -> UnemploymentTesting Consumer Spending "Granger" causes Unemployment rate with a 3 quarter lag (data since 1990)P Value 11.0%

Unemployment -> Consumer SpendingTesting Unemployment rate "Granger" causes Consumer Spending with a 3 quarter lag (data since 1990)P Value 89.8%

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Granger Causality test (data since 1970)

Consumer Spending -> UnemploymentTesting that change in Consumer Spending "Granger" causes Unemployment rate with a 3 quarter lag (data since 1970)P Value 11.4%

Unemployment -> Consumer SpendingTesting that Unemployment rate "Granger" causes changes in Consumer Spending with a 3 quarter lag (data since 1970)P Value 81.8%

We did the exact same exercise using now data since 1970. Surprisingly, the results were nearly as strong as when using the data since 1990 that showed stronger absolute correlation. I suspect that the results using data since 1970 were nearly equally good because even though the difference in absolute correlations (whether A causes B or B causes A) was very small; the difference in actual correlations (including the sign) was nearly as high for the 1970 data than for the 1990 data. Also, the 1970 data had a larger sample which reduces the standard errors, boosts the t Stats, and lowers P values everything else being equal.

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Demonstrating Causality – Granger Causality

• Is the causal association strong? Yes, the correlation of -0.62 is reasonably high especially after demonstrating its [Granger] causality.

• Does it make economics sense? Yes, a change in consumer spending (that accounts for 2/3ds of GDP) has a strong impact on GDP. The business sector reacts accordingly by laying off or hiring people. But, it reacts with a lag.

• Is it temporally correct? Does the cause precede the effect? Yes, the Granger Causality analysis fully demonstrated that.

• Can the causal effect be replicated using different time series? Yes, conducting the analysis using data cut offs of either 1970 or 1990 did not make much difference in “Granger” causation.

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Path Analysis

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Path Analysis: Direct and Indirect Effects

The Correlation of the independent variable can be decomposed into its Direct Effect and Indirect Effect on the dependent variable. The Indirect Effect is derived from intermediary variables in between the independent and dependent one. The Causal Effect is the sum of the mentioned Effects and should equal the Correlation.

Direct Effect

Correlation Causal Effect

Indirect Effect

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Path Analysis – Correlation Table (data since 1990)

Correlation matrix

U lag 3 U lag 2 U lag 1 U lag 0 CS lag 0U lag 3 1.00 0.97 0.88 0.74 -0.62U lag 2 0.97 1.00 0.96 0.87 -0.52U lag 1 0.88 0.96 1.00 0.96 -0.39U lag 0 0.74 0.87 0.96 1.00 -0.22CS lag 0 -0.62 -0.52 -0.39 -0.22 1.00

-0.62-0.52 U lag 2 0.97

CS lag 0 -0.39 U lag 1 0.88 U lag 3

-0.22 U lag 0 0.74

This is Path Analysis starting point. You start with a correlation matrix of all variables. Next, you develop a diagram with the independent variable at the left, intermediary variables in the middle, and the dependent variable in the right. You then embed all the relevant correlations from the table into the diagram. The correlation between Consumer Spending (CS lag 0) and Unemployment lag 3 (U lag 3) (highlighted in yellow) is the relation we tested with Granger Causality. We are looking at the impact of Consumer Spending on all four Unemployment variables (lag 0, lag 1, lag 2, and lag 3). In this case, we treat U lag 0, U lag 1, and U lag 2 as intermediary variable. In turn, those three variables have an impact on U lag 3. This latter impact is just serial or autocorrelation.

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Path Analysis

With standardized variables within a single relationship the Correlation is equal to the Slope.

Correlation = COVAR (X, Y)/(s X)(s Y)Slope = COVAR (X, Y)/VAR XIf both s = 1. Correlation = Slope

In view of the above, the correlations between the independent variable and the intermediary variables are equal to regression coefficients or path coefficients as they are called in Path Analysis.

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Path Analysis – Path CoefficientsRegression Statistics

Multiple R 0.988R Square 0.975Adjusted R Square0.974Standard Error 0.162Observations 76

-0.06-0.52 U lag 2 1.44

CS lag 0 -0.39 U lag 1 -0.45 U lag 3

-0.22 U lag 0 -0.09

Path CoefficientsIntercept -4.6E-16U lag 2 1.44U lag 1 -0.45U lag 0 -0.09CS lag 0 -0.06

The path coefficients between the independent variable (CS lag 0) and the intermediary variables (U lag 0, U lag 1, U lag 2) do not need to be calculated because given that the variables are standardized the slope or path coefficients are equal to the original correlations. We just need to calculate the path coefficients highlighted in yellow. And, those are the regression coefficients shown above generated by a linear regression model including all variables with Unemployment lag 3 quarters as the dependent variable.

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Direct & Indirect Effects

-0.06-0.52 U lag 2 1.44

CS lag 0 -0.39 U lag 1 -0.45 U lag 3

-0.22 U lag 0 -0.09

Decomposing correlations into indirect and direct effectsConsumer spending indirect effect on Unemployment lag 3 A B A x BCS -> Unemployment lag 2 -> Unemployment lag 3 -0.52 1.44 -0.75CS -> Unemployment lag 1 -> Unemployment lag 3 -0.39 -0.45 0.17CS -> Unemployment lag 0 -> Unemployment lag 3 -0.22 -0.09 0.02

-0.56CS direct effect on Unemployment lag 3 -0.06Total causal effect -0.62

Note that the sum of the direct (-0.06) and indirect effects (-0.56) = -0.62 which is exactly the same as the original correlation between change in Consumer Spending and Unemployment rate lag 3 (quarters).

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Path Analysis meaning

Using Path Analysis where the intermediary variables are the lagged variables of the dependent variable may be unusual. Typically, intermediary variables are exogenous; in this case it could be such variables as business investments or government spending.

Nevertheless, in such [endogenous] context Path Analysis does have a meaning. The path coefficient of -0.06 represents the remaining direct effect that consumer spending has on unemployment lag 3 after its effects on all preceding quarters have already been factored in.

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Demonstrating Causality – Path Analysis

• Is the causal association strong? Yes, the sum of the direct and indirect effects at -0.62 is material.

• Does it make sense in an economics context? Yes, as defined previously.

• Is it temporally correct? Does the cause precede the effect? Yes, Path Analysis is most often based on sequential variables.

• Can the causal effect be replicated using different time series? We skipped this question here, but addressed it already with Granger Causality.

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Granger Causality vs Path Analysis

• Granger Causality is a more rigorous causality test than Path Analysis where causality is already inferred by one’s structuring the Path Analysis diagram defining the causal direction of the relationships.

• Path Analysis is more flexible as it accommodates intermediary variables that enrich the analysis.

• To fully demonstrate causality, a narrative test as shown on slides 12 & 20 is required.

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Appendix: Data sources

• Unemployment rate: U.S. Bureau of Labor Statistics.

• Consumer spending: Bureau of Economic Analysis using the time series for Personal Consumption Expenditures.