Questions tagged [time-series]

statistical techniques for application to data whose observations concern an entity or phenomenon at different points in time.

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Deriving ADF unit root test form for the time series with quadratic deterministic trend

I have the following time series process $y_t $ $$\Delta y_t = \delta + \gamma t + \epsilon_t$$ where $e_t$ is white noise process with the variance of $\sigma^2$. I guess that whereas $\Delta y_t$ is ...
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Application of Maximum Likelihood estimation (MLE) to the Generalized Least Square (GLS) model

I have the following regression $$y = X\beta +u$$ where $y$ and $u$ are $(n\times 1)$ and $X$ is a fixed $(n \times k)$ matrix with full column rank and $\beta$ is an unknown $(k\times 1)$ vector of ...
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1 answer
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The effect of stationarity of forecasting

If all that matters in forecasting is getting an accurate forecast then why is using non stationary data a problem. Say you use non stationary data to create a forecast that performs well in a pseudo ...
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How can I say if a variable Granger-causes another variable in a VEC model?

If I have a VEC model output with coefficients and associated standard errors, how I can say if a variable Granger-causes another one by just looking at the output and without any test?
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Is there some standard way to diagnose a structural time series model (also called simple unobserved components model)?

I am dealing with a structural time series model (also called simple unobserved components model), and I wonder if there is some standard way to diagnose this sort of models. In most reference books ...
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How to assess the quality of a forecast?

Let's say I have a time series model (VAR model for example). How can I know that my forecast is good ? I could use the R2 but is there something else? I also know I could just use in sample ...
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1 answer
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Which variable keep/get rid in a time series model?

Let s say I have a big VAR model with many variables. Then I run the model. How can I know which variables I should keep or get rid of if I want to ameliorate my model ? What if my model has so many ...
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can someone help to interpret ACF and PACF graph?

can someone help to interpret ACF and PACF graph? in the graphs, I don't see any decay. am I right about that assumption other than that what can I comment about both? the data plotted is squared ...
1 vote
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Is there granger causality that can be applied to panel data with structural breaks?

I have panel data containing various macroeconomic variables (for example: interest rate, output, unemployment). I would like to run a Granger causality test on some of them, but there are structural ...
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A conceptual question about the limitation of the MA processes

We know that linear time-series techniques are frequently used in macroeconometrics. The Wold Representation Theorem states that any covariance-stationary process may be expressed as an MA process ...
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Proof that a Unit Root process is Difference Stationary

Consider $$y_t =a_1 y_{t-1}+a_2 y_{t-2} +...+a_p y_{t-p} +\varepsilon_t $$ The characteristic polynomial would be: $$(1-a_1L -a_2L^2 -...-a_pL^p) $$ Suppose that there is a unit root, say that $L=1$ ...
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Time series: predicting covariates for missing years from historical values

I'm working with a time-series dataset where my dependent variable is measured annually from 2018-2021 but my covariates are not all measured over the same period. For instance, $x_1$ is measured from ...
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What type of data we use to predict volatility of an asset with GARCH or ARCH models?

When we are trying to feed time-series data to a GARCH or ARCH model, what kind of data should we give the model? A: Absolute difference between daily prices over-time B: % of the difference between ...
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Which econometric model should I use?

I have the following model Expenditures = a + $b_1$ Output + $b_2$ Labor + $b_3$ CapitalStock + $b_4$ D where D is dummy variable that is Categorical. If tax is imposed, it takes 1. Or if tax is not ...
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1 answer
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Derivation of autocovariances Lewis (2021) RES

I am studying this paper, and I don't understand the derivation of the covariances at the bottom of page 3090. Basically I have two shocks: $\varepsilon_{1t}$ has constant volatility $E[\varepsilon_{...
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Modeling asymmetric effects

I was at a colloquium with the Bank of Canada and the presenter was showing some impulse response functions (IRFs) of rate change effects. Intuitively people tend to borrow more when rates decrease ...
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3 votes
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How to combine labor market time series when there is a methodological change that creates a break?

Any suggestions on how to combine time series of labor market variables (such as, for example, labor force participation) when there is a methodological change in the way the variable is calculated by ...
7 votes
2 answers
189 views

Testing for serial correlation with General Regressors

This is from Introductory Econometrics (Wooldridge) 5th Edition page 420. Consider: $$y_t =\beta_0 +\beta_1 x_{1t}+\beta_2 x_{2t}+...+\beta_k x_{kt}+u_t$$ where $Cov(u_t, x_{jt})=0$ for all $j$ but ...
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How to create/replicate Diebold's Canadian Employment Index?

In his textbook "Elements of Forecasting", Francis Diebold presents on page 130 the following series describing it as the Canadian Employment Index, quarterly, seasonally adjusted. The data ...
2 votes
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44 views

Taylor rule estimation with OLS serial autocorrelation

I'm estimating the equation: $$i_{t}=\beta_0+\beta_1\tilde\pi_t+\beta_2\tilde y_t+\varepsilon_t$$ Where $\hat\pi_t=\pi_t-\pi^{target}$ and $\hat y=\ln y_t-\ln y^{\ast}$, are the inflation deviations ...
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1 answer
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What are good (i.e. non-random walk) time series to perform forecasting on?

I have developed a recurrent neural network (RNN) model for time-series forecasting. I now want to test its performance against more standard statistical/econometric models such as ARIMA or VAR. The ...
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Identify time vaying variables in VAR

I am having a panel VAR and would like to identify which varaibles remain constant and which vary over time. I have added time varition in the model. I would like to avoid to use bayesian approarch . ...
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What are claims on a central government?

I downloaded some data from the World Bank's World Development Indicators database. I found a time series called Claims on central government, etc (% of GDP) with code ...
1 vote
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What is difference between Interrupted Time Series (ITS) and Regression Discontinuity in Time(RDiT) analysis?

Can anyone give some details on the difference between Interrupted Time Series (ITS) and Regression Discontinuity in Time(RDiT) analysis? How to choose between them? which one is more robust? Or some ...
1 vote
1 answer
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Cholesky Identification in Structural VAR

Can you suggest me a framework in macroeconomics or finance where identification of a Structural VAR model through Cholesky ordering is still considered credible (in your opinion)? I'm looking for a ...
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Converting from Quarterly to Monthly Data for Output Gap

I am currently working on the quarterly output gap data on FRED. However, I would like to convert the data series into a monthly one. Would an appropriate methodology be to divide the dataset by 3 ...
1 vote
1 answer
33 views

Filling gap in data with correlated series

I have two time series, of different length. A time series is GDP growth. The gdp growth is the series I need, and it is also the longer series, but it has two gaps in two periods one after the other. ...
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1 vote
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Deflating value series in currency A with CPI based on prices in currency B?

Say we have time series data on the annual value of imports in the UK from the US (in US dollars). Would it be appropriate to deflate the latter using the UK Consumer Price Index (which is based on ...
6 votes
3 answers
317 views

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

My question is about the definition of cointegrated. $y_t =y_{t-1}+u_t$ $u_t =\eta_t +0.5\eta_{t-1}$ where $\eta_t\sim N(0,1)$ is i.i.d. white noise. I claim that $y_t$ and $y_{t-1}$ are cointegrated ...
1 vote
1 answer
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Algebra for two period forecasting in AR (3) Model

I wondered if some folks could help fill in a knowledge gap for me with some time-series algebra please regarding the following AR (3): $$x_t = \phi x_{t-1} + \phi_2 x_{t-2} + \phi x_{t-3} + \...
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1 answer
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$I(g)$-terminology

During my studies I came across the $I(g)$-symbol, in the sense that $Z_t \sim I(g)$ for an integer $g$ and $Z_t$ is a times series. What exactly does this mean, and does it have to do with ...
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3 votes
1 answer
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How to recognize correlation in spurious regression case

Assume we are given two independent random walks $$ Y_t = Y_{t-1} + \varepsilon_{1, t}, \quad \varepsilon_{1, t} \sim \mathcal{N}(0, 1) \\ X_t = X_{t-1} + \varepsilon_{2, t}, \quad \varepsilon_{2, t} \...
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1 answer
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Interpreting an ADF test in R

Although being a mathematician, I am fairly new to time series and R. On an assignment I was being asked to check a time series for stationarity in R, only using the $\texttt{adf.test}$ function that ...
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3 votes
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Quantifying treatment effect in multivariate interrupted time series

I have a multivariate time series dataset, from which I am building an ITS (Interrupted Time Series) model by using facebook's Prophet to construct the counterfactual. Let's say I have a y variable ...
0 votes
1 answer
105 views

Stationary vs. Non-stationary data in a BVAR model

I am replicating a paper using BVAR model and I first I have run the model with non-stationary data. Then I just wanted to compare the results with stationary data and I launched the model but I get ...
2 votes
1 answer
36 views

Frequency and time of a time series suddenly changed, are usual methods valid

I have a time series data that used quarterly information on a variable, but has now shifted to bi-monthly releases, from September 2008 to July 2021. This is a survey based measure, released at the ...
3 votes
1 answer
133 views

Durbin Watson Test for an AR(1) process

$(1) y_t =\beta y_{t-1} +\epsilon_t$ $(2) \epsilon_t =\rho \epsilon_{t-1} +v_t$ Where $v_t$ is i.i.d white noise. I know that OLS estimates of (1) are biased. It would then follow that estimates of $...
3 votes
0 answers
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AR(q) Strongly Stationary

Consider an AR(q) process, $u_t$. If the roots of a characteristic polynomial are outside of the unit circle, the AR(q) process is weakly stationary. I've seen this proof that proceeds by showing the ...
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1 answer
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Estimation of variables

I am trying to estimate some variables, but I would like to know if what I am doing is right. I am using some data about labour productivity (in log) and, by using Stata, I filtered (Hodrick-Prescott) ...
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1 vote
1 answer
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Finding consistent but inefficient GMM estimate

Consider the following linear model $$y_t = x_t' \beta +u_t$$ where $t =1,...,T$ and $x_t = (x_{1t} x_{2t} ... x_{kt})'$ , $ \beta$ is $k \times 1$ vector of unknown coefficients, $u_t$ is an iid ...
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2 votes
2 answers
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Intercept in 2nd-stage Error Correction Model (ECM) regression — yes or no?

When doing a two-step ECM regression, do we add an intercept in the 2nd stage regression? I've seen course notes that add an intercept in the ECM, but some do not, so I'm confused if I should include ...
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1 vote
1 answer
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K-step ahead forecast of VAR(2)

I am trying to determine how to write the K-step ahead forecast of a VAR(2), with two variables, as a weighted average of its mean and last observations. I understand that one must use companion form, ...
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IRF vs Autocorrelation function of AR(2)

Will the impulse response function and autocorrelation function of an AR(2) coincide? I say that they will not, because $IRF_{t}(t+k)$ (that is, the effect of a unit shock at time $t$, on the process ...
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2 votes
1 answer
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How to obtain real money balances data

Theoretically real money balances ($m_t$) are defined as: $m_t=\frac{M_t}{P_t}$ Where $M_t$ are nominal money balances, and $P_t$ is the price index of the economy. If I were to make an empirical ...
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1 vote
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Detrending multiplicative model

I have a GDP quarterly series and IND.P monthly series, which I need to detrend. I am following Bernanke et al. paper from 1995 and the way they do it is: ...
0 votes
1 answer
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What does "sample period" in this case mean?

Dasgupta,2019 documented that the treated group comprises all firms that are headquartered in countries that have passed a leniency law by year t. The control group comprises firms in countries that ...
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0 votes
1 answer
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How to handle the missing values' issue of newly listed stocks?

I am trying to test some asset pricing models on 10 portfolios for the period of 2010-2020. The problem is that three of these portfolios included stocks that are newly listed in 2017 and 2018, so I ...
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1 vote
2 answers
62 views

Examples of the use of Vector Autoregressive Models

I am self-learning Vector Autoregressive Models currently, and have practiced on a few datasets. But I wanted to read some actual research papers that use VAR, so that I get an idea of the level of ...
1 vote
0 answers
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What is the importance of moving seasonality in seasonal adjustment?

Both Stable and Moving Seasonality are computed in Sliding span test (or D8 F-Stat test) and M7 statistics. But what is the purpose of finding out the presence of moving seasonality in seasonal ...
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1 vote
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Modelling GDP growth rates with an AR(1) process

I am trying to model the growth rates of real output per worker. I read that GDP growth rates are often assumed to be stationary and are modelled using an AR(1) process, so I took logs and then took ...
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