Key Uses of VAR Models: Forecasting and Granger Causality

Ever feel like you’re trying to predict the unpredictable? Imagine you’re a retailer planning for the next quarter. You need to forecast your sales, but also your advertising spend, and maybe even your competitor’s pricing. The problem is, none of these things exist in a vacuum. Your ad spend (hopefully) boosts your sales, but your sales figures also determine your ad budget for next month. And your competitor’s price cuts directly impact your sales, which in turn affects your ad spend. You’re not just looking at one domino; you’re looking at a whole network of them, all falling at once. How do you forecast this beautiful mess? This is the exact problem that Vector Autoregression (VAR) models were built to solve.

What is a vector autoregression (VAR) model? 🔗

In the world of time series analysis (that is, data collected over time), we often start with simple models. A basic “autoregressive” model, for example, tries to predict a variable’s future value using only its own past values. It’s like saying, “To know where sales will be next month, I’ll look at sales from the last 12 months.”

But as our retailing example showed, this is rarely enough. The real world is interconnected. A Vector Autoregression (VAR) model is a powerful upgrade that treats multiple variables as a single system. Instead of one equation for “sales,” a VAR model builds a small system of equations-one for each variable in your set. In our example, we’d have a system with three equations:

• An equation for Sales, which depends on past sales, past ad spend, and past competitor prices.
• An equation for Ad Spend, which depends on past ad spend, past sales, and past competitor prices.
• An equation for Competitor Price, which depends on past competitor prices, past sales, and past ad spend.

The “vector” part of the name just means a list of variables (e.g., [Sales, Ad Spend, Price]). The “autoregression” part means each variable is “regressed” on (or explained by) its own past values (its “lags”) and the past values of every other variable in the system. It’s the ultimate “it’s all connected” model.

Why not just forecast them separately? 🔗

This is a great question. You *could* build three separate models. But you’d be missing the most important part: the feedback loops. A VAR model captures the dynamic, two-way street of influence. A standard model might tell you that ad spend impacts sales. A VAR model can also capture how sales, in turn, impact the future ad spend, creating a feedback cycle.

By treating all variables as “endogenous” (meaning their values are determined *inside* the model, rather than being an external input), VARs allow us to see how a shock or change in one variable doesn’t just affect another-it ripples through the entire system over time, with echoes and feedback effects that a single-equation model would miss entirely.

The crystal ball: Forecasting with VAR models 🔗

The primary use of VAR models, and what they are famously good at, is forecasting. Because the model captures the rich, interconnected dynamics of a system, it can often produce forecasts that are superior to those from single-equation models or even more complex, theory-based structural models. Economists and central bankers, in particular, love them for this.

How the forecast actually works 🔗

The forecasting process from a VAR model is beautifully logical. The model provides the “best forecast” by working recursively. Here’s a simplified breakdown:

1. Get the “rules”: First, you estimate the model using your historical data. This gives you an “estimated coefficient matrix,” which is just a fancy term for the set of rules the model learned, like “a 1% increase in ad spend *last* month leads to a 0.2% increase in sales *this* month.”
2. Start from today: To forecast the next period (e.g., March), the model takes the most recent available data (the actual, known values from February) and plugs them into the estimated “rules.” This generates a forecast for all variables for March.
3. Keep on rolling: Now, to forecast for April, the model does something clever. It takes the *actual* data from March… oh, wait, that hasn’t happened yet! So, it uses the forecasted values for March that it just generated. It plugs those into the rules to predict April.
4. Recursive projection: It repeats this process over and over. The April forecast is used to predict May, the May forecast to predict June, and so on. This creates a “coherent multi-period outlook” where the forecasts for all variables stay consistent with each other over the entire forecast horizon.

A real-world example: A central bank’s puzzle 🔗

Let’s think about a central bank like the Reserve Bank of India (RBI). Its job is to manage inflation, ensure economic growth, and maintain a stable exchange rate. These three variables-inflation, GDP growth, and the exchange rate-are massively interconnected. High growth might lead to inflation. The RBI’s response (hiking interest rates) will cool inflation but might also slow growth and strengthen the rupee. It’s a complex policy puzzle.

This is a perfect job for a VAR model. In fact, central banks and economists in India use VAR-based models to do exactly this. They use them to create unconditional forecasts (“where are inflation and growth heading if we do nothing?”) and to conduct policy analysis. The model provides a unified framework for seeing how all these key macroeconomic pieces move together, helping policymakers make more informed decisions.

The econometric detective: Understanding Granger causality 🔗

This second major use of VAR models has a slightly intimidating name, but the concept is brilliant and intuitive. It’s called “Granger Causality,” named after its inventor, the Nobel laureate Clive Granger. But the first thing to know about Granger causality is a giant, flashing disclaimer: it is not “causality” in the way you and I use the word.

It does *not* mean that X *causes* Y (in the sense that a billiard ball’s “cause” of movement is being hit by the cue). Instead, Granger causality is a test of predictive relationships.

What does it mean to “Granger-cause”? 🔗

The core idea, developed by Granger, is this: Does the past of variable X contain useful information for predicting the future of variable Y, *even after we’ve already used the past of Y itself*?

Think of it this way: 1. You build a model to forecast your friend’s coffee purchases (Y) using only their past coffee purchases (lags of Y). You get a certain amount of forecast error. 2. Then, you build a *new* model. This one predicts coffee purchases (Y) using both their past coffee purchases (lags of Y) and their past work deadlines (lags of X). 3. If this second model is consistently more accurate (i.e., it has a smaller forecast error), then you can say that work deadlines “Granger-cause” coffee purchases. The past of X had unique, valuable information for predicting Y.

How VAR models test for Granger causality 🔗

This concept fits perfectly into the VAR framework. Remember how a VAR model is just a system of equations? To test if X “Granger-causes” Y, we simply look inside the equation for Y.

That equation for Y includes past values of Y, past values of X, and past values of all other variables in the system. The Granger causality test is a formal statistical test (an F-test) that asks: are the coefficients on the past values (lags) of variable X statistically significant in the equation for variable Y?

• If the answer is yes (the coefficients are not zero), we conclude that X Granger-causes Y. Its past has predictive power.
• If the answer is no (the coefficients are statistically indistinguishable from zero), we conclude that X does not Granger-cause Y. Its past is just noise for predicting Y.

In a VAR system, you can run this test for every pair of variables, building a “map” of predictive influence. Does ad spend Granger-cause sales? Do sales Granger-cause ad spend? (This is called bidirectional causality, or a feedback loop, and it’s very common!).

The classic trap: Prediction is not causation 🔗

Why are economists so fussy about this “prediction, not causation” point? Because of the “lurking variable” problem. The most famous example is that ice cream sales “Granger-cause” shark attacks.

If you run the numbers, you will find, without a doubt, that past ice cream sales are a fantastic predictor of shark attacks a few days later. A VAR model would confirm this predictive link. But does eating a Choco-bar *cause* a shark to attack? Of course not.

The real cause is a third, lurking variable: summer heat. Hot weather causes people to buy more ice cream. Hot weather *also* causes more people to go swimming, which, unfortunately, leads to more shark encounters. The VAR model can’t see the *why*; it only sees the predictive pattern. As Wikipedia’s entry on the topic notes, this is a classic “post hoc ergo propter hoc” (after this, therefore because of this) fallacy. Granger causality is a powerful econometric tool, but it’s a detective’s clue, not a judge’s conviction.

So, what’s the catch? (A few limitations) 🔗

As powerful as they are, VAR models aren’t magic. They come with two main trade-offs that users must be aware of.

Atheoretical: VAR models are famously “atheoretical.” This is both their greatest strength and their biggest weakness. They are not built on any deep economic theory about *how* variables should be related. Instead, they let the data speak for itself. This is great for forecasting (it “often provides superior forecasts” as one text notes), but it makes them less useful for deep theoretical analysis. A VAR can show you *that* a shock to interest rates affects GDP, but it can’t, by itself, tell you *why* (e.g., whether it’s through investment channels, exchange rates, etc.).

The curse of dimensionality: A VAR model’s complexity explodes quickly. If you have 3 variables and you want to include 2 past lags for each, the model has to estimate 21 coefficients. Add a fourth variable, and it jumps to 37 coefficients. Add a fifth, and it’s 56. This “curse” means you need a *lot* of historical data to get reliable estimates, and you’re generally limited to a small number of variables (e.g., 3 to 6) at any one time.

Bringing it all together 🔗

Vector Autoregression models are a cornerstone of modern time series econometrics for a reason. They provide a robust and flexible way to move beyond single-variable analysis and embrace the interconnectedness of the real world. By treating multiple variables as a dynamic system, they give us a powerful engine for forecasting a coherent future outlook and a rigorous framework for testing Granger causality-the hidden predictive relationships that drive our world.

What do you think?

Can you think of a real-world example from your own life or business where two or more variables clearly influence each other in a feedback loop? How could a VAR-like approach (even just the *idea* of it) help you make better predictions?

References

1. https://otexts.com/fpp2/VAR.html
2. https://mse.ac.in/wp-content/uploads/2022/01/Model-1.pdf
3. https://www.aptech.com/blog/introduction-to-granger-causality/
4. https://faculty.washington.edu/ezivot/econ584/notes/varModels.pdf
5. https://en.wikipedia.org/wiki/Granger_causality