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Sunday, February 9, 2014

Primer: Exogenous Versus Endogenous Variables

Figure: introduction.
This primer explains the concept of endogenous variables versus exogenous variables, as those terms are used in economics. Although the distinction between endogenous and exogenous appears simple, there are a lot of subtleties involved when the conversation turns to the real economy and not a particular mathematical model. I illustrate how the same variable can be either exogenous or endogenous, depending upon the needs of the modeller. The example used is critically important to bond investors – the policy rate (e.g., the Fed Funds rate). I also comment on these concepts as used in the analysis of fiscal policy.

The definitions of these terms appear straightforward, and make sense based on the Greek roots exo- (“outside”) and endo- (“proceeding from within”).

An endogenous variable is a variable embedded inside a mathematical model of the economy; hence it is determined by the evolution of all the variables inside the model.

An exogenous variable is a variable that is external to the model; it is chosen in some fashion by the user of the model. Alternatively, it is an input to the model. The choice of exogenous variables will determine the evolution of the variables inside the model.

I will now use the example of the policy rate as either an endogenous or exogenous variable, based on the needs of the modeller.

The Policy Rate As An Exogenous Variable

Figure: Example of exogenous policy rate.
Example where the policy rate is exogenous.
The above diagram shows how the policy rate can be viewed as an exogenous variable. The modeller is assumed to be the Central Bank, which wants to see how it determines the interest rate input variable affects the economy. (Note that there is a complication with this point of view, which I discuss in a section below.)

A mathematical model of the economy is somehow determined, and its evolution is a function of the policy rate (and other economic variables). The Central Bank policymakers decide how they want to set the policy rate, based on the observed economic variables. For example, if the model tells them that inflation will rise too far above target with the current level of the policy rate, they would hike rates. The model is supposed to give them an idea of the sensitivity of the economy with respect to changes in the interest rate, allowing them to calibrate their rate moves.

The Policy Rate As An Endogenous Variable

Figure: example of endogenous policy rate.
Example where the policy rate is endogenous.
The above diagram shows how the situation should be analysed from the point of view of a bond investor. The best method to value bonds relies on rate expectations; the fair value of a bond yield is the expected average of short-term rates until bond maturity. This means that the investor needs a method to determine that expected path of short rates.

This is done by determining the “reaction function” of the Central Bank; i.e., given the observed economic variables, how will the Central Bank react? For example, the investors could replicate the Central Bank’s model (as in the “Exogenous Variable” version), and then use it to determine the path of interest rates to keep inflation at a target level.

The key difference between the bond investor’s and the policymaker’s model is that in the Central Bank’s model, the Central Bank has “free will” with respect to determining the interest rate. It could test various possible rate paths, and see how the model economy reacts. In the bond investor’s model, the Central Bank has no “free will”; it follows a mechanical rule, determined within the model. The exogenous variable is forced inside the model, making it endogenous.

In practice, my guess is that most investors do not directly follow this procedure. What is probably more common is that some form of regression model is run on bond yields versus other economic variables. I would interpret these models as an attempt to approximate the reaction function using statistical methods.

Unfortunately, the bond yield is “floating in space” with no yield curve model relating it to the policy rate. This means that these techniques can miss horribly in unusual situations.

For example, some of these regression-style models showed up in dealer research, and they blew up in the 2010-2011 era. (Probably most of these models were abandoned, or else some new series was datamined-into-place in order to fit the recent data.) Based on some variables like inflation and the ISM, the historical experience might indicate that the 10-year yield should have been something like 5%. This is because the levels of inflation and the ISM were not particularly low relative to their historical averages, and so the models predicted the 10-year should be near its historically average level. However, this does not take into account the fact that there was no plausible trajectory for Fed Funds that:

  1. started out at 0% and would be stuck there for at least a couple of years; and
  2. average 5% over 10 years.

A Complication Regarding Modern DSGE Modelling

A purist would argue that my presentation of the policy rate being exogenous for central banks is outdated. Modern Dynamic Stochastic General Equilibrium (DSGE) models are based on model entities solving optimisation problems over time, and the Central Bank reaction function needs to be embedded in the model (the model entities need the path of interest rates to discount future outcomes). Therefore it appears that the policy rate is also treated as endogenous by policymakers currently.

However, this characterisation of the policy rate being endogenous in the DSGE model only holds true in their original nonlinear form. Economists do not actually solve the original model, rather they sit around philosophising about what they think the solutions are. My analysis is that the true solutions do not look anything like what they suppose.

In any event, Central Bankers do not work with the original nonlinear model. They use linearisations, which are supposed to represent how small deviations around the true trajectory will evolve. They can then use these linearisations to determine how other economic variables react to a step change in the policy rate. This means that, in practice, the policy rate ends up being treated as a variable external to the model*.

Fiscal Policy

In the article "What Is The Primary Fiscal Balance, And Why Its Use Should Be Avoided", I explain the problems with the use of the primary balance in the analysis of fiscal policy. Using the terminology of this article, people are treating the primary fiscal balance as exogenous, when it is endogenous to the economy. (As always, I am discussing here countries that control the currency they borrow in, and that currency is assumed to float freely. A country with a currency peg faces additional constraints on its policies.)

I would treat the following fiscal variables are exogenous:
  • tax rates;
  • per capita welfare and other transfer payments;
  • size of government programme spending.
These variables, and the state of the economy determine the other fiscal variables like the budget deficit. For example, the government can set the income tax rate, but the amount of revenue it will raise depends upon declared incomes, which is a function of the strength of the economy. On the spending side, it can determine the per-person unemployment insurance payments, but it does not know how many people will be unemployed.

This means that the government cannot control the level of the fiscal deficit. This is seen in practice; governments periodically announce deficit targets that they typically miss when the target date arrives.

Stating that tax rates are exogenous implies that the government cannot be forced to change them as the result of economic forces. In particular, this means that the government will not be forced to hike taxes as the result of the mythical "bond vigilantes". However, there are limits on how independent these exogenous variables are; they need to be coherent with each other. For example, it would be difficult to run a welfare state that spends 50% of GDP on government programmes, with government "revenue" at 10% of GDP. (Governments were able to run huge deficits like that in wartime, but they had switched over to a command economy to make that possible.) I discuss the need for taxes to drive demand for currency, a core concept of Chartalism, in this article.

Endogenous Versus Exogenous Money

The debate about whether money is “endogenous” or “exogenous” is a long-standing fight between mainstream economists and the heterodox camps. This is usually where the terms endogenous and exogenous crop up in internet economic debates. This is a very interesting topic, but I will have to defer its discussion to a follow up article as a result of the complexity of the topic.

* Yes, this whole situation makes no sense.

(c) Brian Romanchuk 2014

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