ECOS3021 Business Cycles and Asset Markets University of Sydney
2021 Semester 1
Tutorial #1: Extra Notes on Question 9.
Question 9 of Tutorial 1 asked:
Suppose that the log of quarterly real GDP follows a random walk with drift μ = 0.0488, and shocks that are normally distributed with mean zero and standard deviation σ = 0.05. Suppose real GDP is currently equal to $1. What do you expect the value of real GDP to be next quarter? What do you expect the value of real GDP to be one year from now?
A very good question was asked during the tutorial about whether we need to “correct for the bias in the expectations term given the model is in logs?”. This is a great question, which I want to write a little bit about in this note. Please note I do not expect you to know this for the class, and I will ensure that my questions/answers in future are more clearly worded.
Suppose the log of GDP follows a random walk with drift: ln yt = μ + ln yt−1 + εt
where the shock εt is distributed according to a normal distribution with zero mean: εt ∼ N(0,σ2). In this case, our expectation of εt (i.e. the mean of εt) is zero. If we are computing the expectation of the log of GDP (taking the expectation using information known at time t − 1 – note: the shock, εt, is not known at time t − 1). Then we have:
Et−1[ln yt] = Et−1[μ + ln yt−1 + εt] =μ+lnyt−1 +Et−1[εt]
= μ + ln yt−1 + 0
where we get the second line because the drift is constant, so Et−1[μ] = μ, and GDP last period is known at time t − 1 so Et−1[ln yt−1] = ln yt−1. We get the third line because the mean of the shock is zero. We used these results in the answers to Tutorial 1.
However, these results only work if we are computing the expectation of the log of GDP. Things are more complicated if we want to compute the expectations of the level of GDP. Note that the level of GDP can be written as:
yt =exp(μ+lnyt−1 +εt) = exp(μ)yt−1 exp(εt)
= μˆyt−1εˆt
where we have defined μˆ ≡ exp(μ) and εˆ ≡ exp(ε ). In this case, the shock to the level of GDP, tt
εˆt, is distributed according to a log-normal distribution, with parameters N(0,σ2). You can read more 1
about log-normally distributed variables on Wikipedia: https://en.wikipedia.org/wiki/Log-normal_ distribution. The thing that you should be aware of is that the expectation is no longer equal to zero. Instead:
σε2 E[εˆt] = E[exp(εt)] = exp με + 2
where in Question 9 we assumed that με = 0 and σε = 0.05. So the expectation of the level of GDP is affected by this “bias” in the expectation of the log-normally distributed shock εˆ . So, to compute the
correct expectation for the level of GDP we have:
Et−1[yt] = Et−1[μˆyt−1εˆt]
= μˆyt−1Et−1[εˆt]
σε2
=μˆyt−1exp με+2
And so the correct answer to the first part of Question 9 should correct for this bias, which gives the following:
σ2 0.052
Et−1[yt]=μˆyt−1exp με + 2 =exp(0.0488)×1×exp 0+ 2 =1.05×1×1.0013 =1.0513
which is larger than the answer of 1.05 provided in the solution to Question 9 Tutorial 1.
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