Estimating Dynamic and Distributed Lag Models

When studying economic relationships over time, we often discover that changes in one variable don’t instantly translate into changes in another. Imagine a business deciding to increase its advertising spending-does sales revenue jump immediately, or does the effect ripple through time? Understanding these delayed responses is exactly where distributed lag models become invaluable. But estimating these models correctly requires navigating some tricky statistical waters, particularly when past values of the outcome variable play a role in determining current values.

When can ordinary least squares be your friend? 🔗

The good news is that for simple distributed lag models-those that predict today’s outcome using only past and present values of explanatory variables-ordinary least squares estimation can work quite well. Think of a model where today’s consumption depends on both current income and income from the past few months. As long as your model doesn’t include yesterday’s consumption as a predictor, OLS remains a reliable estimation technique.

Here’s the key distinction: models without lagged dependent variables can be estimated with standard OLS methods. These are sometimes called finite distributed lag models. For example, if you’re studying how crop yields respond to rainfall over several months, and you only use current and past rainfall (not past yields) as predictors, OLS will give you consistent and unbiased estimates, assuming the standard regression assumptions hold.

However, the moment you introduce a lagged dependent variable-say, including last period’s consumption to predict this period’s consumption-the statistical landscape changes dramatically. This is where the lagged dependent variable becomes correlated with the error term by construction, violating a crucial assumption for OLS to work properly. In technical terms, the regressor is no longer strictly exogenous, and OLS estimates become biased and inconsistent.

Why lagged dependent variables complicate everything 🔗

Consider the popular Koyck lag model, which includes one lag of the dependent variable alongside current explanatory variables. This elegant specification allows for an infinite lag distribution while estimating only a few parameters. But there’s a catch: the lagged dependent variable is inherently correlated with past error terms. If your error term exhibits any serial correlation-which is extremely common in time series data-then the lagged dependent variable will be correlated with the current error term too.

This creates what econometricians call an endogeneity problem. Your regressor (the lagged dependent variable) and your error term are dancing together, violating the independence assumption that OLS relies upon. The result? Biased coefficient estimates that don’t converge to the true values even with large samples. This is a serious issue because many economic theories predict dynamic adjustment processes that naturally include lagged dependent variables.

The multicollinearity challenge 🔗

Even when lagged dependent variables aren’t present, distributed lag models face another hurdle: multicollinearity among lagged values of the same variable. If this month’s income is highly correlated with last month’s income (as it usually is), and last month’s income is highly correlated with income from two months ago, then your regression matrix becomes unstable. The variance of your coefficient estimates balloons, making it difficult to identify the separate effects of each lagged variable.

The instrumental variables solution 🔗

So how do we rescue our estimates when lagged dependent variables threaten consistency? One powerful approach involves instrumental variables. The core idea is beautifully simple: find variables that are correlated with your problematic regressor (the lagged dependent variable) but uncorrelated with the error term. These instruments can then be used to extract the variation in your regressor that’s unrelated to the error, yielding consistent estimates.

In the context of distributed lag models, lagged values of the explanatory variables often serve as natural instruments. For instance, income from two periods ago can instrument for last period’s consumption because it occurred in the past (thus predetermined) but still correlates with recent consumption patterns through the economic relationship.

Wallis’s procedure in practice 🔗

One classical instrumental variable approach, known as Wallis’s procedure, specifically addresses models with lagged dependent variables and serially correlated errors. The technique uses lagged independent variables as instruments for the lagged dependent variable. Here’s the intuition: suppose you’re estimating a consumption function where last period’s consumption appears on the right-hand side. You might use income from two and three periods ago as instruments.

The procedure works in stages. First, you regress the lagged dependent variable on your chosen instruments (the lagged independent variables). Then, you use the predicted values from this first stage regression as a substitute for the actual lagged dependent variable in your main equation. This two-stage approach breaks the correlation between the regressor and error term, restoring consistency to your estimates.

However, choosing valid instruments requires careful thought. The instruments must satisfy two conditions: they must be relevant (strongly correlated with the endogenous regressor) and exogenous (uncorrelated with the error term). Recent research has shown that using lagged variables as instruments can be problematic if those lags have direct effects on the dependent variable, so practitioners must carefully justify their instrument choices based on economic theory.

Iterative non-linear least squares for efficiency 🔗

While instrumental variables provide consistent estimates, econometricians often seek something better: efficient estimates that make optimal use of the available data. This is where iterative non-linear least squares methods enter the picture, particularly for models with autoregressive error structures.

The challenge arises because many distributed lag models with lagged dependent variables can be shown to have error terms that follow a moving average process. Unlike observable regressors, these lagged error terms aren’t directly observable, which means we can’t use standard linear least squares. Instead, we need iterative procedures that gradually refine the estimates.

How the iteration works 🔗

Imagine you have a model where the current value depends on lagged values and the error term follows an autoregressive pattern. The iterative procedure begins with initial parameter guesses-often from OLS or instrumental variables estimation. Using these initial values, you construct estimates of the error term, then use these estimated errors to update your parameter estimates. You repeat this process, each time getting closer to the maximum likelihood estimates.

The transformation involved typically accounts for the autoregressive nature of the error term. For instance, if your errors follow a first-order autoregressive process with parameter ρ, you might transform your equation by subtracting ρ times the lagged equation from the current equation. This quasi-differencing removes the serial correlation from the errors, but ρ itself must be estimated, leading to the non-linear optimization problem.

In large samples, these iterative methods yield estimates that are both consistent and asymptotically efficient-they have the smallest possible variance among all consistent estimators. The computational burden has become negligible with modern computing power, making these methods increasingly practical for applied work.

Choosing your estimation strategy 🔗

So which approach should you use? The answer depends on your specific situation. If your model is a simple finite distributed lag without any lagged dependent variables, stick with OLS-it’s simple, reliable, and efficient. Make sure to check your residuals for serial correlation and adjust standard errors if needed.

When your model includes lagged dependent variables, you’ll need to choose between instrumental variables and iterative methods. Instrumental variables are often preferred when you have strong, defensible instruments and smaller samples. The procedure is relatively straightforward and doesn’t require complex optimization algorithms. However, the efficiency loss compared to maximum likelihood methods can be substantial.

Iterative non-linear least squares becomes attractive when you want maximum efficiency and have sufficient sample size for the asymptotic properties to kick in. The trade-off is computational complexity and the need to verify that your algorithm has converged to a global, rather than local, optimum. Modern statistical software packages have made this increasingly automated, but understanding what’s happening under the hood remains important.

Diagnostic checking remains crucial 🔗

Regardless of which estimation method you choose, thorough diagnostic checking is essential. Test your residuals for remaining serial correlation using Breusch-Godfrey or Durbin-Watson tests (though the latter must be interpreted carefully with lagged dependent variables). Check for heteroscedasticity, as it can affect standard errors even when point estimates remain consistent. Consider whether your lag length is appropriate using information criteria like AIC or BIC.

Remember that distributed lag models are powerful tools but require careful specification. Economic theory should guide your choice of which variables to include and how many lags to consider. Statistical tests can help refine these choices, but they can’t substitute for thoughtful model construction based on understanding the underlying economic mechanisms.

What do you think? When working with time series data in your field, have you encountered situations where lagged relationships seemed important? How might you determine whether instrumental variables or iterative methods would be more appropriate for your specific application?

References

1. https://en.wikipedia.org/wiki/Distributed_lag
2. https://www.reed.edu/economics/parker/312/tschapters/S13_Ch_3.pdf
3. https://www.tandfonline.com/doi/full/10.1080/25742558.2018.1538596
4. https://stats.stackexchange.com/questions/3324/why-use-a-lagged-dv-as-an-instrumental-variable
5. https://marcfbellemare.com/wordpress/wp-content/uploads/2019/05/WangBellemareLaggedIVsMay2019.pdf
6. https://en.wikipedia.org/wiki/Moving-average_model
7. https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0228812