E CONOMETR ÍA II: S ERIES DE T IEMPO

W EEK 4.1

Juan R. Hernández1

Licenciatura en Economı́a, Spring 2025

1
I will draw heavily on the contents of the book of reference and some slides in its website.
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Outline

1. Time Series Regression: Dynamic Models

2. Time Series Models

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1. Time Series Regression

Time Series assumptions on the LRM:

A1 The variable x does not contain a single repeated value (and if there
were any other x′ s they would be linearly independent).
A2 The Data Generating Process (DGP) is given by the stochastic process:

ut = yt − f (xt ; θ), E(ut ) = 0, for all t = 1, 2, . . . , n.

Here f (·) is a model and θ is a parameter vector.

A3mi The variable x is mean independent from u, E(u|x) = 0.
A4Ω The error process variance E(uu′ |x) is given by Ω. Allows for
heteroskedasticity and autocorrelation.
A5 The error process u is Gaussian: u ∼ N(0, Ω).

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1. Dynamic Models

We introduced dynamic models. These models are a direct way to

address autocorrelation, but they are interesting in their own right.
They also happen to be an intuitive way to test theories and model
time series.

Change notation with respect to last week:

p1
X p2
X pk
X
yt = α + δj x1,t−j + δj x2,t−j + · · · + δj xk,t−j
j=0 j=0 j=0
py
X
+ γj yt−j + ut
j=1

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1. Dynamic Models

Express the model in first differences. This representation also helps

to model explicitly autocorrelation.
▶ The model would now be
p1
X p2
X pk
X
∆yt = α + δj ∆x1,t−j + δj ∆x2,t−j + · · · + δj ∆xk,t−j
j=0 j=0 j=0
py
X
+ γj ∆yt−j + ut
j=1

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1. Dynamic Models

Express the model in error-correction form. This representation also

helps to model explicitly autocorrelation.
Sometimes the change in y is purported to depend on previous values
of y or xt as well as changes in x:

∆yt = α + β0 yt−1 + β1 x1,t−1 + . . . βk xk,t−1

Xp1 X p2 pk
X
+ δj ∆x1,t−j + δj ∆x2,t−j + · · · + δj ∆xk,t−j
j=0 j=0 j=0
py
X
+ γj ∆yt−j + ut
j=1

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1. Dynamic Models

Dynamic models may have a long run static equilibrium solution,

which is useful to test theories and discipline short-run dynamics.
▶ ‘Equilibrium’ implies that the variables have reached some
steady state and are no longer changing, i.e. if y and x are in
equilibrium, we can say

yt = yt+1 = . . . = y and xt = xt+1 = . . . = xt , and so on.

Consequently, ∆yt = yt − yt−1 = y − y = 0

▶ So the way to obtain a long run static solution is:
(i) Remove all time subscripts from variables.
(ii) Set error terms equal to their expected values, E(ut ) = 0.
(iii) Remove first difference terms altogether.
(iv) Gather terms in x together and gather terms in y together.

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1. Dynamic Models
An example is as follows: Let y be the money demand, and x1 , x2 be
the GDP and short-term interest, respectively.

Preference for liquidity theory suggests there is a long run equilibrium

relationship, called the money demand that should hold. In the short
term, however, deviations are allowed (use zero lags for short term
dynamics):

∆yt = α + β0 yt−1 + β1 x1,t−1 + β2 x2,t−1 + δ0 ∆x1,t + ϕ0 ∆x2,t + ut

then the static solution would be given by

0 = α + β0 yt−1 + β1 x1,t−1 + β2 x2,t−1

β0 yt−1 = −α − β1 x1,t−1 − β2 x2,t−1

α β1 β2
y = − − x1 − x2
β0 β0 β0
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1. Dynamic Models
A second way to get long-run static solution: Start with the ARDL in levels
p1 p2 pk
X X X
yt = α+ δj x1,t−j + δj x2,t−j + · · · + δj xk,t−j
j=0 j=0 j=0
py
X
+ γj yt−j + ut
j=1

Assume long-run (static) form, this is, forget about the time subscript, solve
for y to get:
 
py p1 p2 pk
X X X X
1 − γj  y = α + δj x1 + δj x2 + · · · + δj xk or
j=1 j=0 j=0 j=0
Pp1 Pp2
1 j=0 δj j=0 δj
y = Ppy α+ Ppy x1 + Ppy x2
1− j=1 γj 1− j=1 γj 1− j=1 γj
Ppk
j=0 δj
+ ··· + Ppy xk or
1− j=1 γj

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1. Dynamic Models

As always, there are trade-offs:

▶ We need to relax the assumption that the independent variables
are non-stochastic. (Not too complicated)
▶ Is (a little) harder to interpret: what does an equation with a large
number of lags actually mean?
▶ Note that if there is still autocorrelation in the residuals of a
model including lags, then the OLS estimators will not even be
consistent.
▶ If we want to use the ARDL for forecasting, we need forecasts of
the right-hand-side variables.
Since All ARDL representations are balanced and stationary, we can
use all tests we have seen, and sandwich standard errors.

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1. Dynamic Models

There are a few loose-ends:

▶ How can we mix levels with differences of variables?
▶ How do we choose the lag-length of the ARDL (py , p1 . . . pk )?
▶ What happens if β̂0 = 0 statistically?
▶ How do we obtain more information on the dynamics of the
variables?
We need more structure to answer these questions. Move to Time
Series Models.

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2. Time Series Models
So far we have worked within the family of structural econometric
models, where a dependent variable y is explained independent
variables x. This is used when we aim to study the effects of x on y.

But there are tasks that do not necessarily require to pin-down a

structural relationship. This may be the case when:
▶ We are interested only in the dynamics of a variable.
▶ Is hard to reduce the set of variables that explain said dynamics
(e.g. stock prices).
▶ There are unobservable (non-measurable) variables driving
dynamics (e.g. uncertainty, political risk).
▶ We need to forecast y and we do not have (or need) good
forecasts for x.
In these situation we can build a model for y using only information
contained in their past values. These are known as Time Series
econometric models.
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2. Time Series Models: Stationarity
In this course, we will work with univariate Time Series Models.2
Some notation and definitions are needed before going forward:
▶ We say a time series, yt with t = 1, 2, . . . , ∞ is a time series
process because it depends on time explicitly.
▶ We say yt is a Strictly Stationary Process if

P{yt1 ≤ b1 , . . . , ytn ≤ bn } = P{yt1 +m ≤ b1 , . . . , ytn +m ≤ bn }

▶ We say yt is a Weakly Stationary Process or a

Covariance Stationary Process if

E(yt ) = µ t = 1, 2, . . . , ∞ (1)
2
E(yt − µ)(yt − µ) = σ < ∞ (2)
E(yt1 − µ)(yt2 − µ) = γt2 −t1 all t1 , t2 (3)

2
Almost all concepts translate to multivariate models, such as Vector Autorregressive models. Details are contained in
chapter 7 of the reference book.
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2. Time Series Models: Weak Stationarity
Condition (3) above says that if the process is covariance stationary,
all the variances are the same and all the covariances depend on the
difference (distance in time) between t1 and t2 . The covariance
function is given by

γs = E(yt − E(yt ))(yt−s − E(yt−s )), s = 0, 1, 2, . . .

▶ The covariances, γs , are known as autocovariances.

▶ However, the value of the autocovariances depend on the units of
measurement of yt .
It is thus more convenient to use the autocorrelations which are the
autocovariances normalised by dividing by the variance:
γs
ρs = , s = 0, 1, 2, . . .
γ0
If we plot ρs against s=0,1,2,... then we obtain the autocorrelation
function (ACF) or correlogram.3
3
There exists a statistic call the Partial Autocorrelation Function. It relates the autocorrelation between yt and any given
yt−j . We will not cover this topic however.
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2. Time Series Models: White Noise
A White Noise Process is one with (virtually) no discernible
structure. It is characterized by

E(yt ) = µ
var(yt ) = σ 2
γt−r = σ 2 if t = r, 0 otherwise

▶ The autocorrelation function will be zero apart from a single

peak of 1 at s=0. ρ̂s ∼approx N(0, 1/T) where T = sample size.
▶ We can use this to do significance tests for the autocorrelation
coefficients by constructing a confidence interval.
▶ For example, a 95 % confidence interval would be given by
√
1.96 × 1/ T.
▶ If the sample autocorrelation coefficient, ρ̂s , falls outside this
region for any value of s, then we reject the null hypothesis that
the true value of the coefficient at lag s is zero.
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2. Time Series Models: Hypothesis Testing
We can test the joint hypothesis that all m of the ρk correlation
coefficients are simultaneously equal to zero using the Q-statistic
developed by Box and Pierce:
m
X
Q=T ρ̂2k
k=1

where T=sample size, m=maximum lag length

▶ The Q-statistic is asymptotically distributed as a χ2m .
However, the Box Pierce test has poor small sample properties, so a
variant has been developed, called the Ljung-Box statistic:
m
∗
X ρ̂2k
Q = T(T + 2) ∼ χ2m
T −k
k=1

This statistic is very useful as a portmanteau (general) test of linear

dependence in time series.
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2. Time Series Models: Hypothesis Testing
ACF example: Suppose that a researcher had estimated the first 5
autocorrelation coefficients using a series of length 100 observations,
and found them to be (from 1 to 5):

ρ = (0.207, −0.013, 0.086, 0.005, −0.022)′ .

Test each of the individual coefficient for significance, and use both
the Box-Pierce and Ljung-Box tests to establish whether they are
jointly significant.
Solution:
▶ A coefficient would be significant if it lies outside
(-0.196,+0.196) at the 5% level, so only the first autocorrelation
coefficient is significant.
▶ Q=5.09 and Q*=5.26
▶ Compared with a tabulated χ2 (5)=11.1 at the 5% level, so the 5
coefficients are jointly insignificant.
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2. Time Series Models: Models

Some remarks:
(i) Have you noticed that we have not discussed estimation?
(ii) Have you noticed that we are not using any model yet?
(iii) In what follows we do not discuss estimation.
(iv) We can estimate parameters using LS or MLE.
We will now discuss Moving Average and Autorregressive processes
with their correspondent models. But, what is the difference between
a process and a model?
▶ A process is given by nature. We want to model that process.
▶ A model is a parametrization of a process, here we need to
estimate the parameters.

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