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I've seen it in a few pieces of econometric literature, and googling it turns up lots of papers using it, almost always in reference to state-space models and other dynamic linear regressions.

No papers I've read have defined it, though, and some even refer to it in "quotes" as if they're not sure themselves.

An example from a paper I'm currently making heavy use of:

$$ Y_{t}=\alpha_t+\beta_tX_{t}+\varepsilon_{t} $$

with state equations:

$$ \alpha_{t}=\alpha_{t-1}+\eta_{1t} \\ \beta_{t}=\beta_{t-1}+\eta_{2t} $$ where all errors are independent Gaussian white noise.

Later, in reference to interpreting the Kalman-filtered results, we have the line:

$\alpha(t)$ is a stochastic constant and partials out all systematic influences...[not attributable to $\beta$]

The authors go on to say that it is not statistically different from zero at the 5% level (t-test). My results are congruent with this - after the Kalman filter has stabilised it converges to, and hovers around, zero.

So why "stochastic constant", why not just "constant"?

Example source: International Equity Risk Premia Convergence, pp 18-19.

SLD
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  • those words mean totally opposite things. i don't see how this could make any sense – Zach Mar 01 '13 at 18:47
  • It's a new one on me, and I'm not unfamiliar with state space models. – Glen_b Mar 02 '13 at 00:35
  • I've added the context, which you may have already seen in my previous question @Glen_b. – SLD Mar 02 '13 at 01:24
  • I imagine that the distinction from 'constant' it's because ... while it can't be distinguished from a constant - you certainly can't prove it is a constant. – Glen_b Mar 02 '13 at 02:10

2 Answers2

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$\alpha_t$ takes the role of the "constant" in the observation equation:

$$Y_t = \alpha_t + \beta_t X_t + \varepsilon_t$$

That is, it's the part that doesn't depend on $X_t$.

However, it is not actually a constant, but a stochastic process (i.e. a sequence of random variables, indexed by time), given by the state equation:

$$\alpha_t = \alpha_{t-1} + \eta_{1t}$$

Hence, it is the "stochastic constant" of the dynamic regression model.

It is not a particularly good nomenclature and I have never seen it anywhere else.

Chris Haug
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X is non-stochastic, meaning observations on independent variables are fixed in repeated samples

  • I implies no measurement error in X
  • I implies no serial correlation where a lagged value of Y would be used an independent variable
  • I no simultaneity or endogenous X variable
Mubbasher Munir
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