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10 votes
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Stationarity in Time Series

First, recall that a stochastic process $\{ Y_t \}$ is weakly stationary if : $i)$ The first moment is time independent and finite, i.e. $E(Y_t) \equiv \mu < \infty$ $ii)$ The Variance is time ...
Tony's user avatar
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7 votes

$I(g)$-terminology

It's hard to tell without more context. But mostly sure it means "$Z_t$ is integrated of order $g$", i.e. $\Delta ^g Z_t=(1-d)^gZ_t$ (where $d$ is the lag operator) is stationary, in other ...
manifold's user avatar
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6 votes

Could you recommend a book or lecture notes about time series that is easy to understand?

The book I currently use for my time series class is "Time Series Analysis and Its Applications With R Examples" by Shumway and Stoffer, 4th edition. It's served me fine, and gives lots of ...
Kitsune Cavalry's user avatar
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6 votes
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ARIMA - reason for + MA term

The MA terms are lagged errors (you don’t need to fetch them manually - for example in R you can use Arima function which does this for you and any program/language will have this basic function as ...
1muflon1's user avatar
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6 votes
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Stationarity vs weak dependence

Weak stationarity and weak dependence are complementary conditions. A weakly stationary time series $y_t$ has an underlying statistical process which is time-invariant. This is characterised by three ...
EB3112's user avatar
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6 votes
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Dickey Fuller Test

The problem in the unit root case is that the t-statistic does not follow a t-distribution, not even asymptotically. The issue lies in the distribution of the OLS estimator, $\hat{\phi}$. In the unit ...
Tony's user avatar
  • 1,262
5 votes
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What are accepted econometric methods to find out if a time series is I1 or not?

An I(1) series is also known as a series with a unit root. Therefore, the econometric tests to inquire on the order of integration of a time-series are referred to as "unit-root tests". There are ...
dv_bn's user avatar
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5 votes
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Durbin Watson Test for an AR(1) process

Nerlove and Wallis (1966) result Nerlove and Wallis (1966) have discussed this issue. Their Equation (3) derives the probability limit of the Durbin-Watson statistic as: $$\mathrm{plim}\, d^* = 2 \...
chan1142's user avatar
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5 votes
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How to recognize correlation in spurious regression case

Just regress Y on X: $$Y=b_0+b_1X+ e$$ and you will likely find some negative significant $b_1$ coefficient even though both series are just unrelated random walks. You can also see that as one series ...
1muflon1's user avatar
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5 votes

Does I(1) imply a process is cointegrated with its lag?

You are confusing concept of co-integration with concept of integration. If a series $y_t−y_{t−1}=u_t$ is stationary then series is integrated of order 1 not co-integrated. The term co-integration ...
csilvia's user avatar
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5 votes

Does I(1) imply a process is cointegrated with its lag?

Just to add to the answers already given. I think the easiest way to see that this cannot be the case is using a CVAR formulation. For a reference see Johansen and Juselius (1990) (https://...
Andrew M's user avatar
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5 votes

Testing for serial correlation with General Regressors

My understanding is that the classical test for serial correlation is actually conditional to the validity of the strict exogeneity assumption: $E[u_t|X]=0,$ or, as the requirement applies to any $t$, ...
Bertrand's user avatar
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4 votes

Why Is Cointegration Important In Practice?

I think it is a very classical economic teaching problem - showing how something is relevant in the real world. First, it solved a problem where linear regressions could lead to spurious results: ...
Thorst's user avatar
  • 835
4 votes

Looking for discussion on equilibrium vs dynamic models in econometrics

Economists (most of them) build their models assuming most of the time stochastic dynamic equilibrium. So Economics does not contrast "dynamic" with "equilibrium" - it synthesizes them. It is ...
Alecos Papadopoulos's user avatar
4 votes
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Difference-in-differences with long time horizon and repeated treatments

This question is related to a post I addressed on CrossValidated. The "generalized" difference-in-differences (DiD) estimator is amenable to settings with multiple groups and multiple ...
Thomas Bilach's user avatar
4 votes
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Recursive Substitution in Time Series

\begin{align}y_t &= \alpha + \theta_1y_{t-1}+u_t \\ &= \alpha+\theta_1(\alpha + \theta_1y_{t-2}+u_{t-1}) + u_{t} \\ &= (1+\theta_1) \alpha + \theta_1^2y_{t-2} + \theta_1u_{t-1}+u_{t} \\ &...
D F's user avatar
  • 181
4 votes
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Frequency and time of a time series suddenly changed, are usual methods valid

If you insist on using some standard time series model this will be problematic as standard time series models such as ARIMA require fixed frequency. There are some possible ways of dealing with this: ...
1muflon1's user avatar
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4 votes
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Proof that a Unit Root process is Difference Stationary

It is simple to see once you factorize. In your set up there is only one unit root so the characteristic polynomial can be factorized as: \begin{align} y_t &= a_1 y_{t-1}+a_2 y_{t-2} +...+a_p y_{t-...
Dayne's user avatar
  • 1,745
3 votes

Replicate Romer and Romer (2004) results

Sorry for any confusion in my previous answer but there are 2 steps to this process 1) tracing out the impact on the $\Delta$y's and the lags of the shocks and then 2) accumulating the shocks to get ...
Andrew M's user avatar
  • 426
3 votes
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Interpetation of coefficent in AR(1) model

For a second-order stationary series it is the correlation coefficient between the dependent value and its lag. Specify $$y_{t+1} = a+ \beta y_t + u_{t+1}\qquad u_{t+1}= \text{white noise}$$ The ...
Alecos Papadopoulos's user avatar
3 votes

Conditional variance vs. unconditional variance in ARCH model

I cannot directly answer your question, but I think I can shed some light. What I seem to show is that under some restrictions, the unconditional variance is finite. However, I am not sure how to ...
Elias's user avatar
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3 votes
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Can PPP adjusted values be compared over time?

There are two questions here: Whether it is meaningful to compare PPP-adjusted values intertemporaly Whether the "Big Mac" index is a good index for PPP-adjustment. I will occupy myself ...
Alecos Papadopoulos's user avatar
3 votes

Misspecified autoregressive models

Just think your true model is: $$y_t = \phi_0 + \phi_1 y_{t-1} + \phi_2 y_{t-2} + u_t \tag{1}$$ So, under your true model, $u_t$ is uncorrelated with all explanatory variables. So, instead of ...
Neeraj's user avatar
  • 131
3 votes

Misspecified autoregressive models

As Neeraj has explained correctly, omitting a variable will lead to inconsistent estimates of the included variables. They will be biased both in a finite sample and asymptotically because the ...
Richard Hardy's user avatar
3 votes

Structural VAR and Granger Causality

Every structural VAR (SVAR) model, e.g. $$ B_0 y_t = B_1 y_{t-1} + u_t $$ has an equivalent reduced form (VAR), e.g. \begin{aligned} y_t &= B_0^{-1} B_1 y_{t-1} + B_0^{-1} u_t \\ &= A_1 y_{...
Richard Hardy's user avatar
3 votes

Limit of random walk auto correlation function

Hi: The expression tends to 1.0. The intuition is that, as $t$ gets larger and larger, the $h$ lags that seperate the two processes become more and more negligible and the processes begin to look the ...
mark leeds's user avatar
3 votes

Year Fixed Effects in a Dynamic OLS Regression with Cointegrated Variables

No in pure time series we generally don't use fixed effects. If you have data on lets say monthly frequency you could include dummies for months in general, e.g. having February, March, April ... ...
1muflon1's user avatar
  • 56.9k
3 votes

How much can we trust macroeconometric analysis?

I do not think that macroeconomic analysis would be 'less trust worthy'. In the same ways as you can solve the issue of omitted variables and reverse causality in microeconomics you can do it ...
1muflon1's user avatar
  • 56.9k
3 votes

Regressing (Very) Smooth Time Series

One thing to consider, is that it looks like you may have a unit root, though not necessarily. An example of a unit root would be the stochastic process $y_k=y_{k−1}+\epsilon_{k−1}$, where the error ...
BB King's user avatar
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