# Regressor: Nominal return, continuous return or first difference?

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).

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

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.

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.

• Thank you for your help! Unfortunately, while it is true that it's already in a rate of return form, the interest rate is almost always non-stationary according to an ADF test so it is not an option to include it as is. Also, could you give some detail about your evoluationary algorithm approach? I am thoroughly intrigued! – Jase Dec 16 '12 at 12:41
• Look up Symbolic Regression using Genetic Programming. In essence you are building an expression tree and evolving it to find a model which fits your regression analysis best. Be warned: This might be prone to overfitting, so make sure to leave out some data for testing. – André Christoffer Andersen Dec 23 '12 at 18:16
• -1 unless it's explained why it's fine to put a non-stationary regressor in the linear model. – Jase Dec 28 '12 at 14:11
• I have updated the answer. – André Christoffer Andersen Jan 12 '13 at 21:51
• Thank you. I think because I'm using low frequency monetary policy rates it makes sense for it to be non-stationary. For example if a central bank changes mandate half way through the sample (e.g. adjusts inflation target band from 2-3% to 2.5-4%). Or if the country had hyperinflation in the early 1990s and then finally got it in control by the 2000s. – Jase Jan 16 '13 at 1:02

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. • But the Augmented Dickey Fuller testing shows that it's non-stationary for most countries (p-value is mostly > 0.05). How can I now include it in the regression? Wouldn't I be getting spurious results? Just by eyeballing the data matrix it's obvious that there's trending and all sorts of non-stationary behavior in these series. Completely agree with your second paragraph. – Jase Dec 28 '12 at 8:21 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. • Hi Alexey, it is not clear to me what you're suggesting. Is it possible you could write out the proposed transformation you're suggesting that I use in my model? – Jase Dec 16 '12 at 15:30 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.