Waiting for Consumers to Respond: The Error Correction Model for Long Term Numerial Relationships

There is a clear relationship between income and consumption that is present across time.  Although an increase in income does not necessarily translate into an instant increase in consumption, there is lag between changes in income and corresponding changes in consumption.  Obvious examples are the loss of a job which can reduce income substantially, but it may take a while for some expenditures to change.  Loss of employment does not mean that car, mortgage, and credit card payments go away.  On the other hand, a raise/promotion raises a person’s personal disposable income, but it may take a while before a person actually reacts to this higher income level.  Clearly the relationship between income and consumption is there, but how do lags in consumer response come into the picture?  Do these lags in consumer response to changes in income cause biased estimates in their correlation?

The purpose of this post is to analyze the long term relationship between income and consumption in the United States.  Using monthly data from the Federal Reserve Bank of St. Louis between the periods of 1980 to July 2009, I estimate a pooled regression to identify the relationship between income and consumption.  Then I calculated the Error Correction Model to capture the lag in consumer response to build a better model and thus provide better insight into consumer spending as a function of income.  I find that there is a lag which dampens the consumption behavior of consumers after in increase in personal consumption expenditures.

Testing for Unit Roots (Stationary Time Series) in Income and Consumption

After testing consumption and income at data at the levels one cannot reject the null hypothesis that they are a unit root process.  After taking the first difference of Personal Disposable Income (DPDI) and the second difference of Personal Consumption Expenditure (DDPCE) we can conclude that both series are stationary after those transformations. Recall that a stationary time series is needed for forecasting and hypothesis testing of time series data.

The graph above shows the second difference of Personal Consumption Expenditure which has a ADF test statistic of -13.46

The graph above is the firs difference of Disposable Personal Income is stationary given the ADF statistic of -2.29.

Showing Cointegration of Consumption and Income

We suspect that consumption is dependent on income, but we believe that consumption responds with a lag.  To express this idea formally the following equations will be a helpful introduction into the rigor of the error correction model.

1.  Define a linear combination of the suspected cointegrated variables at time t.

2.  Define a linear combination of a suspected cointegrated variable at time t-1.

3. Write the original model in terms of first differences and include the correction term.  The correction term is the first lag of the error term above.

The error correction model above is a regression on suspected cointegrated time series with the lag of the error term.  This error term captures the short run disturbances between changes in PCE and PDI.

Expected Theoretical Results

What sign should the coefficient in front of the lag error term have?

Case 1:  Positive

The coefficient could be positive if a change in your income, such as losing a job, produces a lagged response in consumption.  You might not be able to adjust instantly so you consume some of your savings or begin to borrow.

Case 2:  Negative

A person gets a raise but does not immediately begin consuming.  The lag error term would serve to reduce the amount of expected consumption because of this behavior of waiting for a secure and permanent source of income.

Empirical Results on EViews

series dpce=pce-pce(-1)
series ddpce=dpce-dpce(-1)
series ddspi=dspi-dspi(-1)
series error=-250.90+.963830*dspi
series u = pce-250.90-.963830*dspi
series lagu=u(-1)

The coefficient on LAGU is negative and statistically significant implying that changes in income don’t reflect automatic changes in consumption.  There is a countercyclical lag in the way consumer behave after increases or decreases in income. Given the rising income in the U.S. the lag in consumer response might be attributed to adjustments in spending after promotions and raises.