[Econometrics] Trends and Detrending

Ignoring the fact two sequences are trending in the same or opposite directions can lead us to falsely conclude that changes in one variable are actually caused by changes in another variable. In many cases, two time series processes appear to be correlated only because they are both trending over time for reasons related to other unobserved factors. 

One popular formulation to capture this trend is

 

yt = α0 + α1*t + et, t = 1, 2,....,

where {et} is an independent, identically distributed(i.i.d.) sequence with E(et) = 0 and Var(et) = σ^2 (σ is the sigma of e).

 

We can write this mathematically by defining the change in et from period t-1 to t as Δet = et - e(t-1).

If Δet = 0, then Δyt = yt - y(t-1) = α1.

 

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If {et} is an i.i.d, sequence, then {yt} is an independent, though not identically, distributed sequence. A more realistic characterization of trending time series allows {et} to be correlated over time, but this does not change the flavor fo a linear time trend. In fact, what is important for regression analysis under the classical linear model assumptions is that E(yt) is linear in t. 

 

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Many economic time series are better approximated by an exponential trend, which follows when a series has the same average growth rate from period to period.  For example, in the early years, we see that the change in imports over each year is relatively small, whereas the change increases as time passes. This is consistent with a constant average growth rate.

An exponential trend in a time series is captured by modeling the natural logarithm of the series as a linear trend (assuming that yt > 0):

log(yt) = β0 + β1t+ et, t = 1, 2,.....

yt itself shows an exponential trend: yt = exp(β0 + β1t+ et)

 

β1 can be interpreted as

Δlog(yt) = log(yt) - log(y(t-1)) ≈ (yt - y(t-1))/y(t-1)

(yt - y(t-1))/y(t-1) is also called the growth rate in y from period t-1 to t. To turn the growth rate tinto a percentage, we simply multiply by 100. 

 

Δlog(yt) = β1 if Δet = 0 for all t

In other words, β1 is approximately the average per period growth rate in yt . 

For example, if t denotes year and β1 = 0.39, then yt grows about 3.9% per year on average.

 

A Detrending Interpretation of Regressions with a Time Trend

 

^yt = ^β0 + ^β1*xt1 + ^β2*xt2 + ^β3*t

 

(Setting ^β3*t as a variable which considers time series changes so that other variables can show their changes regardless of time series changes)

(^β3*t값에 시간에 관한 모든 변화를 통제하게 하여, 다른 변수들은 변수 고유의 변화(시간 변화와 독립되어)만 보여줄 수 있게 함) = detrending

 

We can extend the results on the partialling out interpretation of OLS to show that ^β1, and ^β2 can be obtained as follows.

1) Regress each of yt, xt1, and xt2 on a contant and time trend t and save the residuals.

 

Thus, we can think of  ''y as being linearly detrended. In detrending yt, we have estimated the model

yt = α0 + α1t + et

 

 

the residuals from this regression, ^et = ''yt, have the time trend removed (at least in the sample). 

 

2) Run the regression of ''y on ''xt1, ''xt2

 

(No intercept in necesary, but including an intercept affects nothing: the intercept will be estimated to be zero.) This regression exactly yields ^β1 and ^β2 from ^yt = ^β0 + ^β1*xt1 + ^β2*xt2 + ^β3*t.

This means that the estimated of primary interest, ^β1 + ^β2, can be interpreted as coming from a regression without a time trend, but where we first detrend the dependent variable and all other independent variables.

 

Resource : Jeffrey M. Woolderfige, "Introductory Econometrics : A Modern Approach 5th edition"