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18 votes

Inclusion of lagged dependent variable in regression

I recommend two articles: Achen C. H. (2001) Why lagged dependent variables can suppress the explanatory power of other independent variables (link) Keele, L. and Kelly N. J. (2005) Dynamic models ...
Tony Ladson's user avatar
14 votes

Forecast time series data with external variables

If you fit a model using external variables and want to forecast from this model, you will need (forecasted) future values of the external variables, plain and simple. There is no way around this. ...
Stephan Kolassa's user avatar
14 votes
Accepted

Cross-validation for timeseries data with regression

For the first question, as Richard Hardy points out, there is an excellent blog post on the topic. There is also this post and this post which I have found very helpful. For the second question, ...
Skander H.'s user avatar
9 votes

Can a Variable Be Both Dependent and Independent?

The notation is discussing two different time periods. $y_t$ refers to the GDP at some time $t$. $y_{t+q,b}$ refers to the GDP at some other time. These are measurements of different quantities.
Dave's user avatar
  • 63.3k
8 votes
Accepted

When should we use lag variable in a regression?

When a lagged explanatory variable is used in a model, this represents a situation where the analyst thinks that the explanatory variable might have a statistical relationship with the response, but ...
Ben's user avatar
  • 126k
7 votes
Accepted

How many lags should I include in time series prediction?

Looking at individual autocorrelations may help in simple cases, but this way you could miss lags that are important only jointly but not individually. Alternatively, you may try the following: ...
Richard Hardy's user avatar
7 votes
Accepted

Why does differencing White Noise induce autocorrelation of $-0.5$?

For notational simplicity, let $$X_t \equiv \Delta Y_t.$$ By definition, $$ \text{Corr}(X_t,X_{t-1}) = \frac{\text{Cov}(X_t,X_{t-1})}{\sqrt{\text{Var}(X_t)}\sqrt{\text{Var}(X_{t-1})}}. $$ Here, \begin{...
Richard Hardy's user avatar
6 votes

How to determine $p$ and $q$ in my ARIMA model from these ACF and PACF plots?

As you know, you look for patterns of (partial) autocorrelation coefficients that are large. A rough guide to how large is "large" is to pay attention to any coefficient plotted with a spike ...
whuber's user avatar
  • 325k
6 votes
Accepted

If B is my backshift operator then how do I calculate (1 - B)?

$B$ is not a number or a matrix. It's an operator. You can think of it as a function, or a mapping: it takes a time series and backshifts them, $Bx_t=x_{t-1}$. We could have used functional notation, $...
Stephan Kolassa's user avatar
6 votes
Accepted

Order of integration of a time series process

Hi: $y_{t}$ is I(1) so the first difference of $z_{t}$ is I(1). Therefore, you'd have to difference the difference of $z_{t}$ in order to obtain an I(0) process, so, by definition, it must be I(2). ...
mlofton's user avatar
  • 2,497
5 votes

Understanding lag operators in ARMA models

This is simply what the lag operator does - it operates on the time series by shifting the index one period back - it is defined as $Lx_t\equiv x_{t-1}$. The lag operator behaves very much like a ...
Christoph Hanck's user avatar
5 votes

Choosing the maximum lag length in the augmented Dickey-Fuller test

This is can be a tricky one. These Zivot Notes discuss a slightly more advanced way to select lags for the ADF. That being said, it is good to remember that purpose of including lags is to control ...
Jacob H's user avatar
  • 922
5 votes

Residual autocorrelation versus lagged dependent variable

This boils down to maximum likelihood vs. methods of moments, and finite sample efficiency vs. computational expediency. Using a 'proper' AR(1) process and estimating the parameter $\rho$ (and ...
Thomas Nichols's user avatar
5 votes
Accepted

Selecting lag order for VAR and VECM

Your methodology seems fine. From a theoretical perspective, it broadly agrees with recommendations in time series textbooks. From an empirical perspective, your models have well-behaved residuals, ...
Richard Hardy's user avatar
5 votes

Variance of AR(1) process using lag operator

Assuming $|\phi_1|<1$, you have $ Y_t = (1-\phi_1 L)^{-1} e_t = \sum_{i=0}^{\infty} (\phi_1L)^i e_t = \sum_{i=0}^{\infty} \phi_1^i e_{t-i}. $ Hence, $ Var(Y_t) = \sum_{i=0}^{\infty} \phi_1^{2i} ...
Ale's user avatar
  • 1,670
4 votes

Inclusion of lagged dependent variable in regression

Some say that the inclusion of LDV will biase downward the coefficient of other IVs. To be more specific, using OLS with the inclusion of a LDV can bias your coefficient downwards. Consider the model ...
Ian Barnett's user avatar
4 votes

What to do if ACF or PACF show significant higher lags?

[I believe this is a duplicate - and while I can find questions with this issue explained in comments, the couple that explain it correctly and fully in answers aren't really answering the same ...
Glen_b's user avatar
  • 284k
4 votes

Lag length selection in ARCH-LM test

The ARCH-LM test is a portmanteau test. It tests a number of lags (lag 1 through lag $q$) at once as a group and tells you whether the average ARCH effect within the group is large. The maximum lag $q$...
Richard Hardy's user avatar
4 votes

Why use lag 12 autocorrelation to test January effect?

Let $j_{t}$ be an indicator that month $t$ is January. Imagine the return process is: $$r_t = \mu + b j_t + \epsilon_t$$ Where $\epsilon_t$ is a white noise process. Mean returns are higher in ...
Matthew Gunn's user avatar
  • 22.5k
4 votes
Accepted

Showing the covariance and autocorrelation functions of a stationary time series are symmetric around 0

Take a zero mean process for simplicity. Then, $\gamma_j=E(Y_tY_{t-j})$. Under stationarity, the point in time at which we compute the expectation does not matter. Hence, we may add $j$ to each time ...
Christoph Hanck's user avatar
4 votes
Accepted

How can I use polynomial distributed lag models for longitudinal categorical exposure?

Spline-based distributed lag models are discussed in the book "Longitudinal Data Analysis" by Diggle, Heagerty, Liang, and Zeger. The idea is including one or more lagged covariates in a model to ...
AdamO's user avatar
  • 63k
4 votes
Accepted

If $y_t$ and $x_t$ are cointegrated, then are $y_t$ and $x_{t-d}$ also cointegrated?

To answer your title question: Yes, if $y_t$ and $x_t$ are cointegrated, then $y_t$ and $x_{t-d}$ are also cointegrated. I think you got the intuition right: $y_t$ and $x_t$ are cointegrated and thus ...
Richard Hardy's user avatar
4 votes
Accepted

What does it mean to have only lag significance on certain multiples of 7

We can be confident that this is a day of the week effect. In many places, and perhaps most, what is reported or done or happens as far as Covid is concerned varies with day of the week. This seems to ...
Nick Cox's user avatar
  • 57k
4 votes
Accepted

How can I reduce the propagation of errors in multi-step time series forecasting?

I do not think there is a way around the issue, fundamentally. There are two ways of $h$-step-ahead forecasting: iterative/recursive (like yours) and direct. The latter goes as follows: for $h$-step-...
Richard Hardy's user avatar
4 votes

What possible effects to include when running a Mixed Model, time variable and or lags for DV's and IV's?

You ask important questions, which I address below in turn. 1: My main research question is whether or not there is a relation between 'Health Care' and 'Social Support' how should i decide whether i ...
Erik Ruzek's user avatar
  • 4,940
3 votes

Forecast time series data with external variables

As I see it, you have three options: Use a published forecast for your independent variables or find a model to forecast them. For example, the Census will have forecasted population data. Using the ...
Ezra Boyd's user avatar
  • 161
3 votes

Forecast time series data with external variables

As Yogi Berra said, "It's tough to make predictions, especially about the future." Many stat software modules will generate forecasts based on the univariate stream of time series in the absence of ...
user78229's user avatar
  • 10.7k
3 votes
Accepted

Coefficient bias in ARIMA vs. lagged regression

The difference and/or similarity between the models might be less obvious than you might think. Check out Rob J. Hyndman's blog post "The ARIMAX model muddle". My first attempt was using an ARIMAX(...
Richard Hardy's user avatar
3 votes
Accepted

VAR lag length vs Johansen cointegration test outcome?

This is a usual problem with the two steps procedure, where one selects first the lag, then the cointegration rank depending on the lag chose in the first step. Puzzle 1: The claim that the lag ...
Matifou's user avatar
  • 3,083
3 votes
Accepted

Autocorrelation of concatenated independent AR(1) processes

Executive summary: It seems that you are mistaking noise for true autocorrelation due to a small sample size. You can simply confirm this by increasing the k ...
Candamir's user avatar
  • 1,030

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