Nonparametric regression for locally stationary time series. The time trend deterministically shifts the mean of the time series. [Vogt, 2012] Vogt, M. (2012). differences needed to achieve stationarity, i.e., the The result of the tests is shown below: $$ \begin{array}{c|c|c|c|c|c|c} \textbf{Deterministic} & \bf{\gamma} & \bf{\delta_0} & \bf{\delta_1} & \textbf{Lags} & \bf{5\% \text{CV}} & \bf{1\% \text{CV}} \\ \hline \text{None} & {-0.004} & {} & {} & {8} & {-1.940} & {-2.570} \\ \text{} & {(-1.665)} & {} & {} & {} & {} & {} \\ \hline \text{Constant} & {-0.008} & {0.010} & {} & {4} & {-2.860} & {-3.445} \\ \text{} & {(-1.422)} & {(1.025)} & {} & {} & {} & {} \\ \hline \text{Trend} & {-0.084} & {0.188} & {} & {3} & {-3.420} & {-3.984} \\ \text{} & {(-4.376)} & {(-4.110)} & {} & {} & {} & {} \\ \end{array} $$. general, the hypothesis that a series is against the alternative A polynomial-time trend can be defined as: $$ {\text Y_{\text t}}=\beta_0+\beta_1 \text t+\beta_2 {\text t}^2+⋯+\beta_{\text m} {\text t}^{\text m} \epsilon_{\text t},\text t=1,2,…,\text T $$. To get the gist of this, assume that we are conducting an ADF test on time times with lagged level only: $$ \Delta \text Y_{\text t}=\gamma \text Y_{\text t-1} $$. It should be noted that Determine whether the time series contains a unit root. Now let \(\gamma=(\beta_1-1)\). $$ \text Y_{\text T}=\beta_0+\beta_1 \text T+\epsilon_{\text t} $$, $$ \text Y_{\text T+\text h}=\beta_0+\beta_1 (\text T+\text h)+\epsilon_{\text t+\text h} $$, $$ \begin{align*} \text E_{\text T} (\text Y_{\text T+\text h})&=\text E_{\text T} (\beta_0)+\text E_{\text T} (\beta_1 (\text T+\text h)+\text E_{\text T} (\epsilon_{\text t+\text h}) \\ \Rightarrow \text E_{\text T} (\text Y_{\text T+\text h})&=\beta_0+\beta_1 {(\text T+\text h)} \\ \end{align*} $$. This reiterates the importance of choosing an appropriate model. So, $$ \text E_{\text T} (\text R_{2002} )=-0.25+0.000154×2002=0.058308 $$, $$ \begin{align*} \text E_{\text T} (\text Y_{\text T+\text h} ) & \pm 1.96\sigma \\ & =0.058308 \pm 1.96×0.0245 \\ & =[0.010288,0.106823] \\ \end{align*} $$. $$ \Delta {\text Y}_{\text t}=\gamma {\text Y}_{\text t-1}+(\delta_0+\delta_1 \text t)+(\lambda \Delta {\text Y}_{\text t-1}+\lambda_2 \Delta {\text Y}_{\text t-2}+⋯+\lambda_{\text p} \Delta \text Y_{(\text t-\text p)}) $$, \(\gamma {\text Y}_{\text t-1}\)=Lagged level, \(\delta_0+\delta_1 \text t\)=deterministic terms. From the regression time series equation given, we have \({\hat \beta}_1=0.015\) and \({\hat \beta}_2=0.0000564\) so that the growth rate is given by: $$ {\beta_1+2\beta_2 {\text t}}=0.015+2×0.0000564×240=0.0421\% $$. The distribution of this test statistic has been tabulated as well for the If we subtract \(\text Y_{\text t-1}\) from both sides of the AR(1) above we have: $$ \text Y_{\text t}-\text Y_{\text t-1}=\beta_0+(\beta_1-1) \text Y_{\text t-1}+\epsilon_{\text t} $$. ), then: $$ \begin{align*} (1-\text L)\phi(\text L) \text Y_{\text t} &=\epsilon_{\text t} \\ \phi(\text L)[(1-\text L) \text Y_{\text t}]&=\epsilon_{\text t} \\ \phi(\text L)[(\text Y_{\text t}-\text L \text Y_{\text t} )]&=\epsilon_{\text t} \\ \phi(\text L) \Delta \text Y_{\text t} &=\epsilon_{\text t} \\ \end{align*}$$. If we want to test, in is nonstationary, it does not show a tendency of mean reversion. presence of one unit root. A unit root process does not have a mean-reverting level. reg D.gnp96 L.D.gnp96. If the residuals are not white noise but the time series appears to be stationary, we can include an AR term to make the model’s residuals white noise: $$ \text Y_{\text t}=\beta_0+\beta_1 {\text t}+\delta_1 \text Y_{\text t-1}+\epsilon_{\text t} $$. We can also forecast the value of the time series outside the sample’s period, that is, T+1. From the regression equation, \(\hat \beta_0=5.1062\) and \(\hat \beta_1=0.0443\). However, R's tseries::adf.test() and Python's statsmodels.tsa.stattools.adfuller() give completely different results.. adf.test() shows it's stationary (p < 0.05), while adfuller() shows it's non-stationary (p > 0.05). Describe how to test if a time series contains a unit root. The example is listed below.Running the example plots the sequence of random numbers.It’s a real me… . We are clearly dealing with a non-stationary time series with an upward trend so, if we want to implement a simple AR(1) model we know that we have to perform it on first-differenced series to obtain some sort of stationarity, as seen here. In the statistical analysis of time series, a trend-stationary process is a stochastic process from which an underlying trend (function solely of time) can be removed, leaving a stationary process. More generally, for models involving the The appropriate model for the time series with exponential growth is the Log-linear Trend models. $$ The differenced data will contain one less point than the original data. sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/xegbohtmlnode37.html that the local behavior of this sort of series is independent of The implication of the infinite variance of a random walk is that we are unable to use standard regression analysis on a time series that appears to be a random walk. because economic variables often displays roughly constant growth rates. If the polynomial contains at least If we reject this , time series data is non-stationary, cannot be rejected. In case the deterministic trend term is not significant at 10%, it is then dropped and the constant deterministic term is used instead. For a stationary time series, the ACF will drop to zero relatively quickly, while the ACF of non-stationary data decreases slowly. Seasonality is a feature of a time series in which the data undergoes regular and predictable changes that recur every calendar year. Therefore, we are unable to use the AR model to analyze a time series unless we transform the time series by taking the first difference we get: $$ \text Y_{\text t}=\text Y_{\text t}-\text Y_{(\text t-1)},y_t=\beta_0+\epsilon_{\text t},\forall \beta_0\neq 0 $$, The unit root test involves the application of the random walk concepts to determine whether a time series is nonstationary by focusing on the slope coefficient in a random walk time series with a drift case of AR(1) model. A random walk can be defined as follows: $$\text Y_{\text t}=\text Y_{\text t-1}+\epsilon_{\text t}$$, $$\text Y_{\text t-1}=\text Y_{\text t-2}+\epsilon_{\text t-1}$$. We know that, under log-linear trend models, the predicted trend value is given by: $$ {\text Y_{\text t}}={\text e}^{\hat \beta_0+\hat \beta_1 {\text t}} $$, $$ \Rightarrow {\text Y}_{80}={\text e}^{5.1062 +0.0443×80}=5711.29 {\text { Million}} $$. When In the case that the null of the ADF test cannot be rejected, the series should be differenced and the test is rerun to make sure that the time series is stationary. The approximated parameter are asymptotically normally distributed and hence statistical inference using the t-statistics and the standard error happen only if the residuals \(\epsilon_\text t\) are white noise. Transform the data so that it is stationary. Moreover, let \(\theta(\text L) \epsilon_{\text t}\) be an MA. If we take the natural logarithm on both sides of the above equation we have: $$ \text {ln}\left(\cfrac {\text Y_{\text t+1}}{\text Y_{\text t}} \right)=\text {ln} {\text Y_{\text t+1}}-\text {ln} {\text Y_{\text t}}=\beta_1 $$, $$ \text E(\text{ln }{\text Y_{\text t+1}}-\text {ln} {\text Y_{\text t}})=\beta_1 $$. Online References. model. When the trend is positive, then the growth rate is expected to decrease over time. A series is said to be stationary when the statistical properties (importantly mean, variance and auto-correlation from time series forecasting perspective) of the series is time invariant (i.e. alternative hypothesis including stationarity around a constant and/or a models (4.7). A financial analyst wishes to conduct an ADF test on the log of 20-year real GDP from 1999 to 2019. If we write a time series as the sum of a deterministic mean and a disturbance term (V.I.1-193) (V.I.1-194) where h is an arbitrary function. 12th Jun, 2016. business time series are nonstationary. That is: $$ \cfrac {{\text y_{\text t+1}}-{\text y_{\text t}}}{{\text y_{\text t}}} =\cfrac {{\text y_{\text t+1}}}{{\text y_{\text t}}} -1=\text e^{\beta_1 }-1 $$, Example: Calculating the Trend Value of a Log-linear Trend Time Series. Clearly \(\beta_1=1\) implies that the time series has an undefined mean-reversion level and hence non-stationary. stationarity as well as tests of the unit-root null hypothesis. If I want to forecast this series using ARIMA then what is the difference between forecasting using the original non-stationary series and the forecasting using the now stationary differenced series? equal to its value at time plus a random shock. The Annals of Statistics, 40(5), 2601–2633. In may cases,time series can be thought of being composed of a nonstationary trend component and a zero-mean stationary component. related to the mean of the process, : However, when , the presence of the constant term We present online prediction methods for univariate and multivariate time series that allow us to handle nonstationary artifacts present in most real time series. Judging with our eyes, the time series for gtemp appears non-stationary. Intuitively, if the time series is a random walk, then: $$ \text Y_{\text t}=\text Y_{\text t-1}+\epsilon_{\text t} $$. Just wondering if you are really sure your time series is a) non-stationary and b) can not be brought to a stationary form. Can I search for a time-span that is? the drift, . opposed to linear functions of time. coefficient is not less than one. against the alternative of unit roots. Seasonalities can be modeled using the dummy variables or modeling it period after period changes (such as year after year) in an attempt to remove the seasonal change of the mean. The models presented so far are based on the stationarity assumption, that is, the mean and the variance of the underlying process are constant and the autocovariances depend only on the time lag. Many translated example sentences containing "non-stationary time series" – Japanese-English dictionary and search engine for Japanese translations. Therefore, $$ \Delta \text Y_{\text t}=\beta_0+\gamma \text Y_{\text t-1}+\epsilon_{\text t} $$. An AR(2) model is given by \(\text Y_{\text t}=1.7{\text Y}_{\text t-1}-0.7\text Y_{\text t-2}+\epsilon_{\text t}\). In other words, the time-series is a random walk and hence not covariance stationary. Time Series Analysis Ch 5. There is a large literature on testing for unit roots theory. We can also add the seasonal component (if it exists): $$ \text Y_{\text t}=\beta_0+\beta_1 {\text t}+\sum_{\text j=1}^{\text s-1} \gamma_{\text j} \text D_{\text {jt}}+\delta_1 \text Y_{\text t-1}+\epsilon_{\text t} $$. the slope parameter in the linear deterministic trend model (4.18). and figure 4.11). Cite. However, the binary distinction provided by a hypothesis test can be somewhat blunt when trying to determine the degree of non‐stationarity of a time series. This characteristic makes it difficult to come up with sound statistical inference and model selection when fitting the models. hypothesis testing methodology ensures that the null hypothesis is accepted models (4.25)-(4.26). don't vary with the time). There are two standard ways of addressing it: Assume that the non-stationarity component of the time series is deterministic, and model it explicitly and separately. WSS random processes only require that 1st moment (i.e. SOA – Exam IFM (Investment and Financial Markets). walk model is not covariance stationary because the Thus, trend is a linear function of time. Since period s, all dummy variables are zero, then the mean of the seasonality at time s is: $$ \text E[\text Y_{\text s} ]=\beta_0 $$. I want to check the stationary of a time series data saved in TS.csv.. Is there any problems in the following codes? It is mathematically described as: $$ \text Y_{\text t}=\beta_{0}+\beta_{1} {\text Y}_{\text t-1}+\epsilon_{\text t}$$, $$\text Y_{\text t}=\beta_0+{\text Y}_{\text t-1}+\epsilon_{\text t}$$, Where \(\epsilon_{\text t} \sim \text{WN}(0,\sigma^2) \). Select ‘Graph’. The unit root test is done to sing the Augmented Dickey (ADF) test. Recall that \(\beta_1=1\) implies undefined mean-reversion level and hence non-stationarity. reg D.gnp96 L.D.gnp96. should be relative small, while it should be important in the (V.I.1-195) This can be used to obtain the variance of the transformed series (V.I.1-196) When a time series is not stationary in variance we need a proper The monthly real GDP of a country over 20 years can be modeled by the time series equation given by: $$ \text {RG}_{\text T}=6.75+0.015{\hat \beta}_1+0.0000564{\hat \beta}_2 $$. The quantities in the parenthesis (below the parameters) are the test-statistics. Example: Calculating the Growth Rate of Log-Quadratic Time Trend. After completing this reading, you should be able to: Recall that the stationary time series have means, variance, and autocovariance that are independent of time. You cannot accurately predict any of these results without the ‘time’ component. Unfortunately, most of the economic time series is non-stationary and this fact is often neglected by students and beginning researchers. of the auxiliary regression: The test statistic developed by KPSS is based on the idea that for Clearly, if \(\beta_1=1\), then let \(\gamma=0\). nonstationary behavior as homogeneous nonstationarity, indicating Therefore, in trying to decide wether economic data are stationary or The nonstationary time series include time trends, random walks( also called unit-roots) and seasonalities. Such trends are quadratic as In order to get more insight into the kind of nonstationary In other words, the series Example: Forecasting and Forecasting Confidence Intervals. Let's say that I have a non-stationary time series and that the series can be transformed to a stationary series using a first difference. The Annals of Statistics, 40(5), 2601–2633. Models for Nonstationary Time Series. Therefore, the coefficient of variation is not appropriate to measure in trend series. , we get: Assuming that the random walk started at some time with value , stationarity condition, that is, if some of its roots do not lie The hypothesis statement of the ADF test is: We begin with choosing the appropriate model. in different ways. From the above results, proportional growth in time series over the two consecutive periods is equal. , the process is stationary and the parameter is Intuitively a time series variable is stationary about some equilibrium path if after a shock it tends to return to that path. permanently higher. Assume that the time series is defined as: $$ {\text Y_{\text t}}={\text e}^{\beta_0+\beta_1 \text t},\text t=1,2,…,{\text T} $$. Spurious regression is a type of regression that gives misleading statistical evidence of a linear relationship between independent non-stationary variables. \(\text X\sim \text N(0,\sigma^2)\), then define \(\text W=\text e^{\text X}\sim \text{Log}(\mu,\sigma^2)\). Click on ‘Statistics’ in ribbon of ‘Output’ Window. Intuitively, the confidence intervals for any model can be computed depending on the individual forecast error \(\epsilon_{\text T+\text h}=\text Y_{\text T+\text h}-\text E_{\text T} (\text Y_{\text T+\text h} )\). , for some integer , follows a stationary and business time series. If we substitute \(\text Y_{\text t-1}\) in the first equation, we get, $$ \text Y_{\text t}=(\text Y_{\text t-2}+\epsilon_{\text t-1})+\epsilon_{\text t} $$. We discuss each of the non-stationarities. Therefore, we could not depend on the statistical results if the lag coefficient is greater or equal to 1 (\(|\beta_1 | \ge 1\)). plus drift model, which is a hypothesis that frequently arises in economic The positivity of γ corresponds to an AR time series stationary. Note that the AR component reflects the cyclicity of the time series, \(\gamma_{\text j} \) measures the shifts of the mean from the trend growth, which is \(\beta_1 {\text t}\). If this is repeated (double differenced) and the time series is still non-stationary, then other transformations to the data such as taking natural log(if the time series is always positive) might be required. This means that a non-stationary signal is the kind of signal where time period, frequency are not constant but variable. However, this is a more of an… Many translated example sentences containing "non-stationary time series" – Russian-English dictionary and search engine for Russian translations. Similarly, the value of the time-series at time t is \({\text Y_{\text t}}={\text e}^{\beta_0+\beta_1 \text t}\), and at t+1, we have \(\text Y_{\text t+1}={\text e}^{\beta_0+\beta_1 (\text t+1)}\). A linear trend is defined to be \(\text Y_{\text t}=17.5+0.65{\text t}\). On a graph, a linear trend appears as a straight line angled diagonally up or down. December 17, 2019 in Quantitative Analysis. The Geological Society, pp. The recommended method of choosing appropriate deterministic terms is by including the deterministic terms that are significant at 10% level. Time Series Analysis Ch 5. 4. noise. We have been discussing the random walks without a drift; that the current value is the best predictor of the time series in the next period. This is a problem in time series analysis, but this can be avoided by making sure each of the time series in question is stationary by using methods such as first differencing and log transformation (in case the time series is positive). the In linear time series, the growth is a constant which might pose problems in economic and financial time series. This test is popularly known as the Dickey-Fuller Test. Consider the following quarterly time series with deterministic seasonalities and non-zero growth rate: $$ \text Y_{\text t}=\beta_0+\beta_1 \text t+\gamma_1 \text D_{1 \text t}+\gamma_2 \text D_{2{\text t}}+\gamma_{3} \text D_{3\text t}+\epsilon_{\text t} $$. Okay, I hope this is clear by now and that I can proceed (only a bit more theory left). procedure. If is a stationary polynomial, we say If the time-series originates from an AR(1) model, then the time-series is covariance stationary if the absolute value of the lag coefficient \(\beta_1\) is less than 1. If we want to check the stationarity of a time series or a linear What is the 95% confidence interval for the second year if the forecasting residual errors (residuals) is a Gaussian white noise? special case in which the slope parameter of model (4.30) is . to as the deterministic trend term. When the non-stationarity in the series is only caused by variation in the trend over the time. processes). estat bgodfrey . constant term to the random walk model: The random walk with drift is a model that on average grows each period by hypothesis of trend stationarity versus a stochastic trend be shown to correspond to the coefficient of in the However, many series do not behave in that way. Ex. By considering the models at all levels allows us to choose the best model when the time series are highly persistent. This is exactly the kind of problem illustrated by the baseball attendance/Botswana GDP example ~ 119 ~ Show the Granger-Newbold results/tables Dickey-Fuller tests for unit roots The models presented so far are based on the stationarity assumption, that we get: The random walk with drift model results of adding a nonzero This drift parameter plays the same role as Thus, the unit root process can be described as: $$ \psi(\text L) \text Y_{\text t}=\theta( \text L) \epsilon_{\text t} $$, $$ (1-\text L)\phi (\text L)=\theta (\text L) \epsilon_{\text t} $$, Example: Checking for Unit Roots using the Lag Polynomials. Neither of these conditions are met in this test. can be upward or downward, steep or not, and exponential or This shows one way to make a non-stationary time series stationary — compute the differences between consecutive observations. The null hypothesis of trend and processes are by far the most important cases for The model is given by: $$ \text Y_{\text t}=\beta_0+\sum_{\text j=1}^{\text s-1} \gamma_{\text j} \text D_{\text {jt}} +\epsilon_{\text t} $$. That is, \(|\beta_1 | < 1\). . As we have seen the properties of a time series depend on its In fact, it Quadratic trend models can potentially capture nonlinearities such behavior: it wanders up and down randomly with no tendency to return to any But many economic and The model started in April 2019; for example, \(\text y_{(\text T+1)}\) refers to May 2019. In particular, their Data Generating Process may include nuisance Click on ‘Line plots’. When the lag coefficient is precisely equal to 1, then the time series is said to have a unit root. If we subtract \(\text Y_{\text t-1}\) on both sides we get: $$ \begin{align*} \text Y_{\text t}-\text Y_{\text t-1}&=\text Y_{\text t-1}-\text Y_{\text t-1}+\epsilon_{\text t} \\ \Rightarrow \Delta {\text Y}_{\text t}&=0×\text Y_{\text t-1}+\epsilon_{\text t} \\ \end{align*} $$. Judging with our eyes, the time series for … For instance, the Ljung-Box statistics may suggest rejection of the null hypothesis. | ISBN: 9780125649100 | Kostenloser Versand für alle Bücher mit Versand und Verkauf duch Amazon. Assuming that there are s seasons in a year. figure 4.11). that the three models considered (4.24), (4.25) A mortgage analyst produced a model to predict housing starts (given in thousands) within California in the US. Transformations such as logarithms can help to stabilize the variance of a time series. substracting to both sides of the equation: Up to now it has been shown how to test the null hypothesis of a random Thus, let us assume that the variance of the process is: Thus, the transformation must be chosen so that: More generally, to stabilize the variance, we can use the power With such a trending pattern, a time series good survey may be found in Dickey and Bell and Miller (1986), among used to describe the behavior of finance time series such as stock The appropriate method of selecting the lagged differences is the AIC (which selects a relatively larger model as compared to BIC). Using the lag polynomials, let \(\Delta \text Y_{\text t}=\text Y_{\text t}-\text Y_{(\text t-1)}\) where \(\text Y_{\text t}\) has a unit root (implying that \(\text Y_{\text t}-\text Y_{(\text t-1)}\) does not have a unit root. Taking expectations in (4.20) given the past information the random walk with drift model has no particular trend to which it for some , shocks have completely permanent effects; a unit shock time series, the nonstationarity leads to the presence of unit Some time back, two good fellas named David Dickey and Wayne Fuller have developed a test for stationarity. Select ‘Time series’. Specifically, we show that applying appropriate transformations to such time series can lead to improved theoretical and empirical prediction performance. Seasonal differencing is done by subtracting the value in the same period in the previous year to remove the deterministic seasonalities, the unit roots and the time trends. Then click on ‘… Then, the test statistic for the null Kwiatkowski, Phillips, Schmidt and Shin (1992) (KPSS) have developed a test for the unless there is strong evidence against it, it is not surprising that a It is given by: $$ \text {ln }{\text Y_{\text t}}=\beta_0+\beta_1 \text t+\beta_2 {\text t}^2 $$. Solution … The random walk equation is a particular case of an AR(1) model with \(\beta_0=0\) and \(\beta_1=1\). In simpler terms, when observed across any regular time intervals they will remain the same. hypothesis of one unit root against the alternative of stationarity $$ \text Y_{\text t}=\beta_0+\beta_1 {\text t}+\epsilon_{\text t} $$. What is the trend estimated value of the sales in the 80th week? and (4.26) cover all the possibilities under the null transformation introduced by Box and Cox (1964): One of the dominant features of many economic and business time , and tabulated the corresponding critical values. If a shock increases the value of a random walk, there is While implementing the ADF test, it requires that a choice of deterministic terms and number of the lagged differences to be included. Other alternatives such as residual diagnostics, can be useful. [Nason, 2006] Nason, GP 2006, Stationary and non-stationary time series. Seasonal effects are observed within a calendar year, e.g., spikes in sales over Christmas, while cyclical effects span time periods shorter or longer than one calendar year, e.g., spikes in sales due to low unemployment rates. In other words, if there is a unit root in an AR(1) model (with the dependent variable being the difference between the time series and independent variable of the first lag) then, \(\gamma=0\) implying that the series has a unit root and nonstationary. This is known as differencing. . Practically speaking, the polynomial-time trends are only limited to the linear (discussed above) and the quadratic (second degree) time trend. Nonstationarity in the mean, that is a non constant level, can be modelled The estimated parameters are \(\hat \gamma_1=6.25,\hat \gamma_2=50.52,\hat \gamma_3=10.25\) and \(\hat \beta_0=-10.42\) using the data up to the end of 2019. the parameter plays very different roles for and . Therefore, \(\gamma=0\) is the test for \(\beta_1=1\). In a quadratic time trend, the parameter can be estimated using the OLS. economic and business time series, arising series much less To be stationary a time series requires three things: Constant mean across all t. Constant variance across all t. Autocovariance between two observations is only dependent on distance between the observations, which we will refer to as the lag h. Real data usually doesn’t usually meet these standards unless we are measuring something such as white noise. critical values for the corresponding statistics, denoted by However, this is not In a random walk, time series depends on each other and their respective shocks. 2. variable increases at an increasing or decreasing rate. integrated process of order 1, . Graphical representation of data helps understand it better. What does the model predict for March 2020? introduces a deterministic linear trend in the process (see graph (b) in Sometimes the linear trend models result in uncorrelated errors. (Note that for the first time period t=2000 and the last time period is t=2020). Also recall that the mean of a log-normal distribution is given by: $$ \text E(\text W)=\text e^{\mu+\frac {\sigma^2}{2}} $$. , be the LS residuals frequently. outside the unit circle. KPSS derived the So that: $$ \Rightarrow Δ_4 \text Y_{\text t} =\beta_1 (\text t-(\text t-4))+\epsilon_{\text t}-\epsilon_{\text t-4 } $$, $$\Delta_4 \text Y_{\text t}=4\beta_1+\epsilon_{\text t}-\epsilon_{\text t-4} $$.
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