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, ...
- 12.1k
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.
- 67.1k
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 ...
- 133k
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{...
- 69.5k
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 ...
- 334k
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, $...
- 131k
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). ...
- 2,865
6
votes
How to determine the optimal lag length in time series?
If you are mainly interested in forecasting, then please do add that tag. A lag length that is "optimal" for forecasting accuracy is not necessarily the same as an "optimal" lag in ...
- 131k
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, ...
- 69.5k
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} ...
- 1,690
5
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-...
- 69.5k
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 ...
- 121
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 ...
- 23k
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 ...
- 34.8k
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 ...
- 64.8k
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 ...
- 69.5k
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 ...
- 59.4k
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 ...
- 5,880
3
votes
Accepted
Cointegration with lagged variables
Suppose you have two $I(1)$ series, $y_{1,t}$ and $y_{2,t}$. They can be decomposed into
\begin{aligned}
y_{1,t} &= x_{1,t} + s_{1,t}, \\
y_{2,t} &= x_{2,t} + s_{2,t}; \\
\end{aligned}
where $...
- 69.5k
3
votes
Accepted
How many lags for ADF test based on ACF, PACF
You can choose not to provide lags, and let AIC or BIC decide the lags for the ADF test.
ACF/PACF won't tell you much about the number of lags to put in for the ADF test. PACF being cut off after 1 ...
- 857
3
votes
Accepted
Augmented Dickey-Fuller test performed regression in package urca. Problem with lags
Perhaps when setting the maximum lag to $k$, all models are estimated on a sample where $k$ initial observations are not considered (but they are used when lags of the original time series are formed),...
- 69.5k
3
votes
"Zero-lag" for ARMA model
This is the representation as a lag polynomial and not as regression coefficients, i.e. standard ARMA(p, q) can be represented as
A(L) $y_t$ = B(L) $u_t$
where L is the lag operator and A(L) is the ...
- 3,312
3
votes
Lag length selection in a VAR model: information criteria favour zero lag
Based on the criteria (lowest AIC, lowest BIC, etc.), zero lag is preferred, which means a model with just an intercept but no lags. From such a model you will not be able to obtain impulse response ...
- 69.5k
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 ...
- 1,060
3
votes
Does LSTM Eliminate Need for Input Lags?
I believe it is actually pretty clear what you mean by the term input lags, but I will state explicitly.
When doing a regression problem with an LSTM, a input signal $ \mathbf{x} \in \mathbb{R}^{n \...
- 865
3
votes
Inclusion of lagged dependent variable in regression
Yes, you should be wary of Nickell bias in a small T large N situation (Nickell, S. (1981). Biases in dynamic models with fixed effects. Econometrica: Journal of the Econometric Society, 1417-1426.)
...
- 245
3
votes
Accepted
Can I use dynlm without any lagged variables?
You can use the dynlm() command without specifying a dynamic regression (no lagged dependent variable as regressor). However, what you get then as output is just a ...
- 1,497
3
votes
Lag between Forecast and Actual Value
Your model seems to be generating a noisy naive forecast i.e.:
$\hat{Y}_{t+1} = Y_t + Noise(t)$,
I don't think RF is a good approach for this.
You might want to try a random walk model which ...
- 12.1k
3
votes
Time series regression - Lags of independent variable
Concerning the significance of your model:
I think the problem lays in your number of observations. If I understood you correctly you have a time series of 15 data points. If you now include five lags ...
- 107
3
votes
Are there limitations to backshift operator algebra in Time Series Analysis?
I wonder about the model.
Here's why. Let's assume (as is implied) that $\theta_w$ and $\theta_x$ are nonzero. Notice that
$$\theta_w x_t - \theta_x w_t = \theta_w(\theta_x x_{t-1} + \theta_x u_{t-...
- 334k
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