Regressor: Nominal return, continuous return or first difference?

Question

Suppose the application is linear models in financial econometrics. If we want to analyze stocks, the standard approach is to take the continuous/log return: $\ln{ \frac{P_t}{P_{t-1}} }$. Suppose, however, that I want to include interest rates ($=:I_t$) as a regressor in my explanatory framework. How should I construct my regressor?

• $\ln{ \frac{I_t}{I_{t-1}} }$

• $\frac{I_t}{I_{t-1}} - 1$

• $I_t - I_{t-1}$

They all result in $I(0)$ series at the $\alpha = 0.03$ level according to Augmented Dickey Fuller testing. The frequency is either daily (overnight rates) or monthly (monetary policy rates).

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Edit: The variable $I_t$ is non-stationary (for monthly frequency) for most countries.

Answer 1

See edit and comments, this response might not be applicable to the question:

When performing regression you would tend to want your regressors to be of similar type, or at the very least range. Assuming you use log return for price changes I would recommend using the untransformed interest rate. The reason for this is that they are the same type of entity, rate of returns.

$R_t = \ln\frac{P_t}{P_{t-1}}$

$R_{t+1} = \theta_0 + \theta_1R_{t} + \theta_2I_{t} + \epsilon$

You can of course use more fancy transformations, but this would be the natural starting point. Personally I use an evolutionary algorithm to evolve the regressor transformations.

Don't worry about the interest rate being always positive. If this matters at all it will be pushed in to the intercept weight.

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Edit:

Given that the interest rate and data resolution you are looking at displays tendencies to be non-stationary I would retract my recommendation above. However, this does make me wonder if you have enough data since I would intuitively expect interest rates to not trend in the long run.

In your shoes I might have attempted to try evolutionary symbolic regression to transform the interest rate data, as discussed in the comment section. When doing this you could try to use your ADF test results as a fitness measure. The resulting transformation function can be used prior to your linear regression model. Remember to split in to test and training datasets in order to detect overfitting.

Answer 2

Economically, the interest rate should be stationary. Unlike a price series, where a price of $10 may not have had the same meaning for a given stock many years ago as it does today, an interest rate of 10% always means the same thing. Hence I side with Andre's earlier answer that you should use the untransformed interest rate.

Also, you need to think more about the hypothesis you are trying to test, and whether it should depend on the level or the change (however you represent that change). Your model should not just be about trying to pick up some statistical correlation. Think about the purpose of running this regression.

Answer 3

I guess you want your regressor to be a market invariant. The invariants are market variables that can be modeled as the realization of a set of independent and identically distributed random variables at least over the investment horizon.

For equities, the invariant is the compound return. For fxed-income, changes in yield to maturity are considered as invariants. Thus you should use them in your regression. See the book "Risk and Asset Allocation" by Meucci, chapter 3.

Answer 4

I'm thinking the first difference is the most sensible. If we take the Ho-Lee short-rate model for example:

$$dr(t) = \theta(t)dt + \sigma dW(t).$$

Taking the log-return or the continuous-return doesn't admit a very nice or intuitive representation, because we get:

$d(\ln r(t)) = r^{-1}(t)dr(t) - \frac12 r^{-2}(t)d\langle r\rangle(t)$

I expect a similarly messy expression would hold for the nominal returns. However taking the simple difference that I proposed is quite nice:

$\Delta r(t) = \theta(t)\Delta t + \sigma \Delta W(t)$

where $W(t)-W(t-1)\overset{d}{=}W(1) \overset{iid}{\sim} \mathcal{WN}(0,1)$.

This would apply to the other short-rate models. Of course I'm assuming $\Delta (\ln r(t))$ doesn't simplify to something nice, which I haven't gone through.

Also, there is also nothing wrong econometrically as the series is $I(0)$ with a size $\alpha = 0.03$ test for all countries in the sample. At least that's where I'm at now.