Overview "Time series analysis" is an umbrella term for a wide variety of procedures which analyze data over time. For instance, any repeated measures design in any procedure is a form of time series analysis. However, this section primarily treats ARIMA (Auto-Regressive Integrated Moving Average) and other models which fit the pattern of a single variable (ex., stock prices, budget totals, votes) over time. When one series is seen as a function of one or more other data series (ex., student grades over time as a function of teacher resources over time) it is now common to use procedures associated with linear mixed models (LMM). When one series is seen as a function of one or more other data series but the focus is on time-to-event (ex., time-to-death after alternative treatments, time-to-adoption of an innovation in relation to changing state resources over time), it is common to use a parametric event history procedure or the semiparametric Cox regression procedure. These are discussed separately. longitudinal and growth curve modeling in LMM latent growth curve modeling in SEM See: event history analysis Cox regression Of the procedures discussed in this section, exponential smoothing creates a data-driven best fit line showing data trends. Curve-fitting tests whether a best fit line may be created based on any of a number of common models, such as linear, exponential, logistic curve, or others. Interrupted time series (ITS) models determine if an event affects the slope or intercept of before-even versus after-event data points. ARIMA models, often used for large economic time series, are typically used to determine if present data points can be predicted from past data points (ex., to predict future sales based on past sales). ARIMA intervention models use ARIMA techniques for purposes similar to ITS modeling.

Contents Key concepts and terms Exponential smoothing Regression models (curve-fitting) Interrupted time series models ARIMA models ARIMA intervention models Assumptions Frequently asked questions Bibliography




• Key Terms and Concepts

  • Time series designs
    • Simple time series design. The usual time series design is simply the collection of quantitative observations at regular intervals through repeated surveys, such as unemployment indexes collected by the Bureau of Labor Statistics. Analysis of such series involves a number of special statistical considerations such as dealing with the problem of autocorrelation, discussed below.

    • Cohort analysis design. The cohort design involves study of a group which has experienced a major event, such as those who were of draft age during the Vietnam War. Thus, a cohort is a set of people who experienced an event at roughly the same time. A standard cohort table is constructed in which the columns are the times the data are collected and the rows are the groups (usually age groups, such as 21-30, 31-40, etc) from whom data are collected. Ideally, the data collection time interval (the survey interval) is the same as the age span of the groups (the cohort interval) being studied. In a standard cohort table, with means of the variable of interest in the cells, inter-cohort cross-sectional comparisons for different age cohorts at the same time period are read down the columns, inter-cohort cross-period comparisons for comparable cohorts over time are read across the rows, and intra-cohort trends for the same age group over time are read diagonally from left to right and top to bottom.

    • Panel studies design. Panel studies are similar to cohort analyses except the same individuals are interviewed in each period, whereas in cohort studies only a random sample of the same age group is interviewed. In panel designs, such as those built into the National Election Studies, individuals serve as their own controls across survey periods, whereas in cohort analyses, random sampling is assumed to serve a control function, assuring similarity of a cohort by gender, race, etc., as it is measured in successive time periods. Where cohort designs measure net change, panel designs measure gross change (for instance, many more people may switch party identification in a given period than are measured by the net effect of all such switches). Normally panel designs cannot measure net change for an entire population because only certain birth cohorts are studied as they evolve over time, and new cohorts entering the population are not studied. Unlike cohort studies, panel studies have the special problem of sensitization: subjects may react differently on a second survey simply because they have had the experience of the first one. To check for a sensitization effect, panel studies may require control groups matched to the panel groups, further increasing the great expense of this design. Panel data regression is discussed in the section on multiple regression.

    • Event-history design. This is a type of panel study in which the periods of observation are not arbitrarily spaced but instead is taken at each stage of a sequence of events. The timing and spacing of observations thus becomes a critical variable in its own right. Moreover, often in event history studies the data are not interviews of individuals but rather measurements pertaining to organizations or even governments. Event history analysis is covered more fully in a separate section.

  • Time series effects. There are three types of time-series effects, having to do with age, period, and cohort. Disentangling these three types of effects for one set of time-series data is a major challenge of time series analysis, discussed below.
    • Age effects are effects related to aging or the life-cycle. For instance, individuals often tend to become more conservative as they age.

    • Period effects are effects affecting all cohorts in a given historical period. For instance, individuals who experienced the Great Depression became more likely to support social welfare policies.

    • Cohort effects are effects which reflect the unique reaction of a cohort to an historical event, or which were experienced uniquely by the cohort. For instance, the post-WWII cohort which reached draft age during the Vietnam War experienced unique issues which seem to be associated with increased alienation from government.

  • Dependence in a time series refers to serial dependence -- that is, the correlation of observations of one variable at one point in time with observations of the same variable at prior time points. It is the object of many forms of time series analysis to identify the type of dependency which exists, then to create mathematical formulae which emulate the dependence, and only then to proceed with forecasting or policy analysis.

  • Stationarity occurs in a time series when the mean value of the series remains constant over the time series. Frequently, differencing (see below) is needed to achieve stationarity. A stricter definition of stationarity also requires that the variance remain homogenous for the series. Sometimes this can be achieved by taking the logarithm of the data. Because many time series (most economic indicators, for instance) tend to rise, simple application of regression methods to time series encounters spurious correlations and even multicollinearity. A first step in time series analysis is to achieve stationarity in the data to avoid these problems. Stationarity is discussed further under "Assumptions."

  • Differencing is a data pre-processing step which attempts to de-trend data to control autocorrelation and achieve stationarity by subtracting each datum in a series from its predecessor. Single differencing is used to de-trend linear trends. Double differencing is used to de-trend quadratic trends. Differencing will drive autocorrelation toward 0 or even in a negative direction. As a rule of thumb, if single differencing yields autocorrelation spike in an ACF plot (discussed below) >.5, then there has been over-differencing. Over-differencing is also indicated by a residual plot where there is a change in sign from one time period to the next. As another rule of thumb, the optimal level of differencing will be the one with the lowest standard deviation. Sometimes adding moving average (q) terms (see below) to the model will compensate for moderate over-differencing. Note that with each degree of differencing, the time series is shortened by one.

  • Specification. Specification may involve testing for linear vs. nonlinear dependence, followed by specifying either linear models (AR (auto-regressive), MA (moving average), ARMA (combined), or ARIMA (combined, integrated)) or nonlinear models (TAR (threshold autoregressive), Bilinear, EXPAR (exponential autoregressive), ARCH (auto-regressive conditional heterescedastic), GARCH (generalized ARCH)). The procedures in SPSS that handle autoregressive models are AREG (for AR(1) models) and ARIMA (for more general models), which are found in the SPSS Trends module. ARIMA is also found in SPSS under Analyze, Time Series, ARIMA.

  • Autocorrelation is the serial correlation of error terms for estimates of a time series variable, resulting from the fact that the value of a datum in time t in the series is dependent on the value of that datum in time t - 1 (or some higher lag). Autocorrelation can be detected visually by examining a regression line scatterplot. Distances from the regression line to the actual values represent error. When no autocorrelation exists, dots representing actual values will be randomly scattered around the full length of the series. Negative autocorrelation exists when, as one moves along the x axis, the next observation tends to be lower than the previous one, then the next one higher, then lower, and so on, indicating a negative or reactive dependency exists for observations in the series. A positive autocorrelation extis when, as one moves along the x axis, there are series of above-the-line observations, then series of below-the-line observations, then more highs, etc., indicating that a positive or inertial dependency exists for the observations.
    • The Durbin-Watson coefficient is a test for autocorrelation. That is, it tests the time series assumption that error terms are uncorrelated. The Durbin-Watson coefficient tests only first-order autocorrelation (lag t-1). The value of d ranges from 0 to 4. A value of 2 indicates no autocorrelation; 0 indicates positive autocorrelation; and 4 indicates negative autocorrelation. For a given level of significance such as .05, there is an upper and a lower d value limit. If the computed Durbin-Watson d value for a given series is more than the upper limit for the case of positive serial correlation, the null hypothesis is not rejected. If the computed d value is less than the lower limit, the null hypothesis is rejected. If the computed value is in between the two limits, the result is inconclusive. For the case of negative first-order serial correlation d must be more than (4 - d-lower-limit) to reject the null hypothesis. In SPSS, one can obtain the Durbin-Watson coefficient for a set of residuals by opening the syntax window and running the command, FIT RES_1, assuming the residual variable is named RES_1. Click here for further explanation of the Durbin-Watson coefficient.

    • Box-Ljung test. The Box-Ljung test tests the significance of autocorrelation at each lag. ITs significance value should be less than or equal to .05 for all or almost all lags. SPSS supports this test.

    • Pankrantz criterion. The Pankrantz criterion states that the autocorrelation divided by its standard error must be less than 1.25 for the first three lags and less than 1.60 for subsequent lags, to conclude the series has no significant autocorrelation. This criterion is applied to the series residuals to establish that error is not autocorrelated.

    • Generalized least squares estimation (GLS). When autocorrelation is present, one may choose to use generalized least-squares (GLS) estimation rather than the usual ordinary least-squares (OLS). In iteration 0 of GLS, the estimated OLS residuals are used to estimate the error covariance matrix. Then in iteration 1, GLS estimation minimizes the sum of squares of the residuals weighted by the inverse of the sample covariance matrix.

  • Decomposition refers to separating a time series into trend, cyclical, and irregular effects. Decomposition may be linked to de-trending and de-seasonalizing data so as to leave only irregular effects, which are the main focus of time series analysis.

  • Model order refers to the model's lag order. A lag order of -1 is common, indicating variables on the right hand side are lagged one time unit (that is, the variable on the left hand side of the equation -- the variable being forecast -- is matched with values of the independents one time period previous. Testing for model order follows identification and specification, and is the first step in estimation. Model order tests include the likelihood ratio test (LR), the final prediction error test (FPE), and the autoregressive transfer function test (CAT), to name a few.

  
  

• Exponential Smoothing

  Exponential smoothing is a form of time series analysis in which the researcher is interested less in modeling than in sheer prediction of the next period based on data about the current and recent periods. It is used for short-term, "one step ahead" forecasting in practical situations, and is not appropriate for long or even medium-term forecasting.

  Weighting. A key question in forming an exponential smoothing line is how much to count the current period, the previous period, and earlier periods when estimating the value in the next period. Exponential smoothing software allows up to four weighting parameters to be assigned:

  1. Alpha: This parameter affects level estimates and varies from 0 (old observations count just as much as the current observation) to 1 (the current observation is used exclusively).

  2. Gamma: This parameter affects trend estimates and varies from 0 (trends are based on all observations counting equally) to 1 (the trend is only based on the most recent observations in the series). Gamma is only used if there is a trend in the series.

  3. Phi: This parameter affects damping estimates and is used instead of gamma when it is thought that the series is dying out. A value of phi near .1 uses all observations to estimate any trend toward dying out, whereas phi values near .9 respond rapidly to any recent observations indicating the trend is dying.

  4. Delta: This parameter affects seasonality estimates and is used only when the data shows seasonal cycles. Delta values near 0 cause all points to count equally while delta values near 1 estimate seasonality primarily from recent observations.

  Selecting which parameters are needed. The researcher begins by considering which parameters need to be set. Only alpha needs to be set if there is no seasonality, the trend is not dying (damping), and the there is no trend (the series varies randomly about its mean). For alpha to apply, there does need to be autocorrelation (memory), meaning that adjacent points are not random but tend to be relatively close together, even if there is no overall trend. Autocorrelation, trends, damping, and seasonality can all be assessed by looking at a sequence plot of the time series, obtained in SPSS by selecting Graphs, Sequence. (The Sequence Plot dialog box also provides for logarithmic transformations, differencing, and seasonality differencing).

  Grid search for needed parameters. To help estimate smoothing parameters, SPSS provides a grid search function. One chooses Analyze, Time Series, Exponential Smoothing from the menus, then in the Exponential Smoothing dialog box. The grid search function causes SPSS to create a sequence of equally spaced values for alpha and for each value calculates a measure of how well the predictions agreed with the actual values. The parameters that produce the smallest SSE (sum square of errors) are the best-fitting parameters. By default, SPSS displays the 10 best-fitting sets of parameters and their corresponding SSE values. (Warning: if you are estimating more than one parameter, the size of the grid grows exponentially). After the grid search comes up with the optimal parameter setting, SPSS adds two new series to your file. The series fit_1 contains the predicted values from the exponential smoothing, and err_1 contains the errors. Select Graphs, Sequence, from the SPSS menus to obtain a sequence plot of the new smoothed series fit_1. The original series amount and the fit_1 forecasts are both show, and their correspondence indicates the degree to which the exponential smoothing forecasts are tracking actual values.

  Models. SPSS provides exponential smoothing in the menu system under Analyze, Time Series, Exponential Smoothing. The Exponential Smoothing dialog box allows the user to select from four models:

  1. Simple. The series has no overall trend and shows no seasonal variation.

  2. Holt. The series has a linear trend but shows no seasonal variation.

  3. Winters. The series has a linear trend and shows multiplicative seasonal variation.You cannot select this option unless you have defined seasonality with the Define Dates command.

  4. Custom. You can specify the form of the trend component and the way in which the seasonal component is applied.

  Residual analysis. The SPSS exponential smoothing procedure automatically adds the variable err_1 variable to the file. This series is simply the difference between the actual value and the prediction value. It can be plotted by selecting Graphs. Sequence from the SPSS menus. This plot is inspected to assure that residuals are randomly distributed. A finding of non-randomness indicates the model is inadequate.

  
  

• Regression Models (Curve-Fitting)

  Regression models follow the strategy of dividing the time series into two parts. The regression model is developed for the estimation dataset (also called the historical or training dataset). Then it is validated by running it on the hold-out dataset (also called the validation dataset).

  Inspecting the series. Selecting Graphs, Sequence, from the SPSS menu creates a plot of the series. By inspecting the series, the researcher gains a rough impression of whether it would be reasonable to think that some sort of curve might be fitted to the pattern displayed.

  Fitting curves. Select Analyze, Regression, Curve Estimation from the SPSS menus. In the Curve Estimation dialog box's "Models" section, one may check the type of curve wanted: linear, power, quadratic, cubic, inverse, logistic, exponential, or other. Only one independent is allowed. Click the Save button in the dialog to save predicted and/or residual values for each model, for purposes of later comparison. Note that the Save button allows the researcher to specify the range of observations to be predicted.

  Output. For each model selected, SPSS output will show these parameters:
  

  Dependent = The variable name of the dependent variable
  Mth = The type of model, such as QUA for quadratic or CUB for cubic
  Rsq = The coefficient of multiple determination, representing the percent of variance in the dependent explained by the model
  d.f. = degrees of freedom
  F = The F value for the model
  Sigf = The significance value for the model as a whole, based on F
  b0 = the constant
  b1 = the linear coefficient
  b2 = the quadratic coefficient
  b3 = the cubic coefficient
  

  Thus the formula for a quadratic model will be Dependent = b1*case + b2*case-squared + b0, where case is the sequential case number (representing the time variable).

  Validation. While selecting the model with the highest R-squared is tempting, it is not the recommended method. For instance, a cubic model will always have a higher R-squared than a quadratic model. The recommended method for selecting which model is best is cross-validation. That is, the formulas for each model based on the estimation dataset are applied to the hold-out dataset, then the R-squares are compared based on output for the hold-out dataset. Alternatively, the determination may be made graphically by overlaying sequence plots of both models for the hold-out dataset.

  Leading Indicator Regression. While simple curve-fitting uses time (the sequential case number) as the predictor variable, in some settings one or more leading indicators may be available. A leading indicator, of course, is a variable whose value in the present period is a good predictor of the dependent variable in a future period.

  Obtaining stationarity.The researcher can determine if the dataset has leading indicators by using the cross-correlation function (CCF) discussed below. However, CCF can only be used if the time series is stationary. In the common situation where all variables are increasing over time, cross correlation will be spurious: all variables will appear to be leading indicators of all other variables simply because all are increasing. Stationarity of a time series is achieved through differencing, which is subtracting the previous value from the current value. First-order differencing subtracts once and removes linear trends. Second-order differencing subtracts twice and removes exponential trends. Not all series can be rendered stationary by differencing. Differencing can be done on the fly inside the CCF option in SPSS.

  Using cross-correlation to identify leading indicators and lags. CCF is found in the SPSS menu system under Graphs, Time Series, Cross-Correlations. The Cross-Correlations dialog box allows the research to treat any or all time series variables and to apply first or second order differencing (or higher, but that is rarely done), as well as apply natural log transforms. One might apply cross-correlation to a suspected leading indicator and to the dependent variable. Upon clicking OK, Cross-Correlations will yield a cross-correlation plot in which the x axis is lags. The lag with the greatest correlation will show having the highest bar. To put it another way, a good leading indicator will have a high bar on one of the positive lags (1 lag ahead or greater).

  Creating the leading indicator variable. After a leading indicator is found and the optimal number of lags ahead it predicts is determined, the next step is to create a new variable which for any given time period contains the value of the indicator from the proper number of lags ago. In SPSS Trends, select Transform, Create Time Series. In the Create Time Series dialog box, move the indictor variable into the variables list. Then highlight the contents of the Name text box and type a name that you want to replace it. Then choose the Lag function from the Function drop-down list. The Order text box shows a value of 1. Highlight this and replace it with a higher lag value if CCF so indicated. Click Change. The New Variables list will now contain something like "leadvar=LAG(inquiries,3)", for the case where the newly created leading indicator variable "leadvar" is the "inquiries" variable with a lag of 3 time periods. Click OK to create the new time series.(Note that since there is a lag of 3 in this example, the first three observations will have a period, representing a missing value, since the file lacks information about the index prior to observation 1. Other observations will equal the value of "inquiries" three rows higher.)

  Linear regression. Select Analyze, Regression, Linear, from the SPSS Base menu system. Follow normal regression procedures to specify the dependent variable and to make the new leading indicator variable the independent. If cross-validation is to be used (recommended, see below), regress only the evaluation cases, saving a hold-out portion of the time series for validation. Note that time series regression frequently violates the regression assumption of uncorrelated errors. When this happens, the significance levels and goodness-of-fit statistics reported by Linear Regression are unreliable. Nonetheless, one can still use the regression equation to make forecasts on the basis of a leading indicator. The regression coefficients themselves are not biased by the autocorrelated errors.

  Cross-validation. To apply the linear regression model to all observations in the time series, including the hold-out data, from the SPSS menu select, Transform, Compute. In the Computer Variable dialog box, let the Target Variable be a new variable such as "predict". In the Numeric Expression text box enter the regression formula computed in the linear regression step above. It will take the form such as "32+ 1.54*leadvar", where "32" is the constant, "1.54" is the b coefficient for leadvar, and "leadvar" is the leading indicator, which will be the lagged version of some other variable (of "inquiries" in the example above). Click OK to create the new variable "predict". Go to Data, Select Cases, and select All Cases. For graphical cross-validation, obtain a sequence plot for "predict" and the "dependent" variable, to visually inspect how well "predict" tracks the dependent not only for the evaluation cases but also for the hold-out validation observations.

  Autoregression. SPSS Trends contains an Autoregression procedure which will correct for correlated error and allow proper interpretation of significance and R-squared statistics associated with time series regression models. In the SPSS menu system, autoregression is found under Analyze, Time Series, Autoregression. This opens the Autoregression dialog box, where one may select a dependent variable and one (or more) independent variables. The researcher then selects among three autoregression models:
  1. Exact maximum-likelihood. A newer model which can handle missing data. It also can handle the case where one of the independent variables is the lagged dependent variable.

  2. Cochran-Orcutt. A widely used model for first-order autogression when there are no imbedded missing values.

  3. Prais-Winsten. This is a generalized least-squares (GLS) method. It cannot be used when a series contains imbedded missing values.

  In the Autoregression dialog box, the researcher can also specify whether to include a constant in model (the default). and whether to save predicted values and residuals. The Autoregression procedure can create five new variables: fitted (predicted) values, residuals, the standard errors of the prediction, and the lower and upper confidence limits of the prediction.

  The Autoregression procedure displays final parameters and goodness-of-fit statistics. The b coefficients and their significance in an autoregression model (after autocorrelation has been removed) may be compared with the corresponding coefficients in a simple regression model. Independents which were shown to be weak or insignificant in a simple regression model may be revealed to be significant in an autoregression model. The parameter estimates in an autoregression model are much more likely to represent the "true" relationships since correlated errors are taken into accoount. Autoregression is discussed further in Chapter 9 of the SPSS Trends manual.

  
  

• Interrupted Time Series Models

  • ITS design. With interrupted time series (ITS) designs, in simple form the researcher has a number of data points before and after an event of interest. The objective of ITS is to determine if the event has a significant effect. There are two major types of effect: the dependent mean shifts at the event point (the intercept changes), and/or the rate of change over time shifts at the event point (the slope changes). It is possible to have an intercept effect without a slope effect. For instance, one could have a zero slope before and after an event which shifts the mean value of the dependent upward abruptly. Effects may be abrupt (an immediate jump in the slope or intercept at the event point) or gradual (the slope and/or intercept begins to shift at or with a lag after the event point) Effects may also be permanent (post-event observations continue to differ from pre-event) or temporary (post-event observations show an effect, but the effect reverts to the pre-event pattern).
    • Comparison groups. ITS design may be improved by having observations for a comparison group which does not experience the event. In complex designs, one may have multiple events.

    • Validity. ITS designs must defend against selection bias as well as other threats to validity such as maturation, history, and regression toward the mean (see the section of "Statnotes" on internal validity).

  • Procedures. A number of different techniques exist to evaluate ITS effects, not the least of which is simple graphical inspection of the time series. Statistically, if there the number of observations is large enough (say 50 or more), then conventional time series techniques such as ARIMA may be used (see section below on ARIMA intervention models). Often, however, the number of data points is small and interrupted time series regression is more appropriate. Other techniques such as partial correlation are sometimes used as well.
    • Testing for heteroscedasticity. In ITS regression, the main focus is usually on comparing slopes before and after the event. Comparison of regression coefficients (slopes) requires homogeneity of the variances upon which the compared coefficients are based. Testing for heteroscedasticity is a first step necessary to justify ITS regression. White's test see White, 1980) may be used. SPSS does not support White's test directly but a macro exists for its implementation. In White's test, if the null hypothesis is rejected, homoscedasticity cannot be assumed. There are other tests also, such as Goldfeld-Quandt.

    • Correcting for autocorrelation. Further, one must test to see if autocorrelation is present in the time series (usually the case) and if so, one must correct the slope estimates for serially correlated residuals. The Durbin-Watson test for autocorrelation is standard. If autocorrelation is present, slope estimates must be corrected using the Prais-Winsten (aka Yule-Walker) method. This is a form of generalized least squares estimation (see Johnston, 1984) and is the default in the SPSS AREG procedure, which is the autoregression option in the SPSS Trends module.

  • Output. After testing for heteroscedasticity and correcting for autocorrelation, the actual ITS regression output presents the significance of the slope for the pre-event period, the significance of the slope for the post-event period, and the significance of the change in slopes from the pre-event to post-event sets of observations. In addition, one could also test for the significance of the difference in intercepts, which is a type of difference of means test.

  • Software. The main statistical tools used for ITS analysis are the SPSS Trends module and AREG procedure, SAS PROC AUTOREG, and the Stata xtreg procedure.

  
  

• ARIMA Models

  • ARIMA models are Auto-Regressive Integrated Moving Average models, also called Box-Jenkins models. ARIMA models predict a variable's present values from its past values. They are sometimes called "p,d,q models" because three parameters must be specified in the SPSS Analyze, Time Series, ARIMA procedure (or similar in other packages):

  • ARIMA modeling involves three stages: (1) Identification of the initial p, d, and q parameters, using autocorrelation and partial autocorrelation methods; (2) Estimation of the p (auto-regressive) and q (moving average) components to see if they contribute significantly to the model or if one or the other should be dropped; and (3) Diagnosis of the residuals to see if they are random and normally distributed, indicating a good model. An ARIMA (0,1,1) model means no autoregressive component, differencing one time to remove linear trends, and a lag 1 moving average component. Lags: Note that lag n means an effect n weeks back, so a spike in the data every four weeks would be one of lag 3, not 4.

  • Identification of ARIMA parameters:

    • Autoregressive component (p). Usually 0, 1, or 2, A value of p=0 means the raw data have no autocorrelation, p=1 means current observations of the series are correlated with themselves at lag 1 (the most common situation), p=2 means correlation at lag 2 also, and so on. An autoregressive component of p = 2 thus means that the dependent (the time series value) is affected by the preceding two values, x_t-1 and x_t-2 independently.

    • Integrated component (d). Usually 0, 1, or 2. The d (integrated) component is simply 0 if the raw data are stationary to begin with, 1 if there is a linear trend (the usual case), or 2 if there is a quadratic trend. Higher positive values are possible but very rarely useful. An ARIMA (0,1,0) model is a random walk model in which differencing can be used to remove the linear trend but the remaining variation cannot be explained either on an autoregressive nor on a moving average basis.
      • ARFIMA models are fractional ARIMA models which allow d to be other than integer. Where in ARIMA, the researcher specifies d à priori, in ARFIMA d is estimated using maximum likelihood methods. The (p,d,q) order of an ARFIMA model is selected by comparing AIC statistics for alternative models. Point estimates of d, called the "fractional integration parameter," may more accurately reflect the effect (persistence) of the given independent variable and also serve to compare measures of the variable across samples. For instance, Ray (1993) found that modeling an ARFIMA process as an ARIMA autoregressive process led to serious errors of forecasting when d is near 0.5. However, ARFIMA requires large samples. Simulations by Crato and Ray (1995) demonstrated that a large number of observations was necessary to obtain a significant advantage in using ARFIMA models. ARFIMA is supported by the popular RATS time series program from Estima.

    • Moving average component (q). Usually 0, 1, or 2. A value of q=0 means there are no shocks in the series and the series is purely an autoregressive one. A setting of q=1 means current observations are correlated with shocks at lag 1, q=2 means they are correlated with shocks at lag 2, and so on. Normally the researcher will set either p or q to a positive value but not both as that may cause overfitting the solution to noise in the data.

      The values of the p and q parameters may be inferred by looking at autocorrelation and partial autocorrelation functions as discussed below.

    • Constants: When d=0 there is usually a constant term equal to the mean of the time series. When d=1 there is usually a constant term reflecting the non-zero average trend. When d=2 there is normally no constant; if a constant is added, then it reflects the value of the "trend in the trend."

    • There can also be components reflecting continuous variables entered as independents in the ARIMA model.

    Autocorrelation and partial autocorrelation functions (ACF and PACF) can also be used to estimate p and q. Specifically, ACF and PACF plots plot deviations from zero autocorrelation by time period: the larger the positive or negative autocorrelation for a period, the longer the plot line to the right (positive) or left (negative) of zero. ACF and PACF are obtained in SPSS under Graphs/Time Series/Autocorrelations.

    • Autoregressive models. AR models are indicated when PACF cuts off sharply at lag x but ACF declines slowly. To determine tentatively the value of p, look at the PACF plot and determine the highest lag at which the PACF is significant.

    • Moving average models. MA models are indicated by a rapidly declining ACF and PACF. If the ACF does not decline slowly but rather cuts off sharply at lag x, this is suggests setting q=x, thereby adding a moving average component. If autocorrelation is negative at lag-1 then this also indicates the need for an MA (q) term higher than 0.

    Other rules of thumb:

    • ARIMA (p,0,0): ACF is spiked at lag p and declines toward 0. PACF is spiked at lag 1 to lag p.
    • ARIMA (0,1,0): Random walk model. The only effect is a non-seasonal differencing to remove a linear trend. ACF is either constant or is balanced between positive and negative. PACF is spiked only at lag 1.
    • ARIMA (1,1,0): First-order autoregressive model. There is non-seasonal differencing to remove a linear trend, and lagging the dependent variable by 1.
    • ARIMA (0,1,1): Simple exponential smoothing model. There is non-seasonal differencing to remove a linear trend, and lagging shock effects by 1.
    • ARIMA (0,0,p): ACF is spiked at lags 1 to p, declining sharply thereafter to 0. PACF is spiked at lags 1 to p, declining more slowly toward 0.
    • ARIMA (p,0,q): ACF and PACF both decline slowly toward 0. PACF declines erratically due to shock effects.
    • ARIMA (1,1,1): A mixed model. Warning: normally one does not include both autoregressive effects and moving average effects in the same model because this may lead to overfitting the data to noise, and may reduce the reliability of interpretation of the significance of individual components in the model.
    As a rule of thumb, one may wish to start with p=1 and/or q=1 and then increase the p and/or q values if the ACF and PACF for the residuals display spiking.

  • Seasonal effects: If there are spikes in the data every four periods for quarterly data, or every 12 periods for monthly data, there is a seasonal effect. In SPSS, prior to asking for the ARIMA model, one asks for seasonal differencing by Data, Define Dates, Cases Are: (and put in the time period, such as "years, months" or "years, quarters"), which creates variables like YEAR_, MONTH_, and DATE_. Then in Graphs, Time Series, Autocorrelations, put in the difference you want (typically 1) and the seasonal differencing you want (typically 1, for monthly data with a periodicity of 12). Alternatively, in Analyze, Time Series, ARIMA, simply enter the p, d, and q parameters you want for both the simple differencing column and the seasonal differencing column in the "Model" portion of the ARIMA dialog box.
    A seasonal ARIMA effect is indicated by a second parenthetical term with a subscript indicating the periodicity. ARIMA (1,1,0)(1,1,0)₁₂ thus refers to a lag 1 autoregressive model with single differencing to remove a linear trend, and no moving average (lagged shock) effects, with a seasonal effect at lag 12.

  • Significance testing of auto-regressive and moving average components and of independent variables: On asking for ARIMA in SPSS, the output will include a t-test in the "Variables in the Model" section for AR (ex., AR1 for autoregressive component with p=1), MA (ex, MA1 for the moving average component with q=1), and if seasonal effects are specified, the SAR and SMA components. The probability levels of the t-tests for these components, as well as for any independent variables, reveals which significantly contribute to the model. Non-significant components and variables are normally dropped. This is akin to significance testing of b coefficients in ordinary regression.
    • Unit roots. If the coefficient of the AR(1) coefficient approaches 1 (has a "unit root"), this means the autoregressive component is emulating a first difference. This would lead the researcher to remove the autoregressive (p) component and add the order of differencing (d+1) in its place. In a higher-order AR(n) models, a unit root is when the sum of AR coefficients approaches 1. One would then reduce p by 1 and increase d by 1. Likewise, if the MA(1) coefficient approaches 1 then the MA(1) component is cancelling a first difference, leading the researcher to remove the MA(1) (i.e., q) term and reduce differencing (d) by 1. In MA(n) models, if the sum of MA coefficients approaches 1, then q is reduced by 1 and the order of differencing is reduced by 1.

  • ACF, PACF, and model fit. After fitting the model, ideally there will be no significant ACF's and PACF's. While such perfection is not common in practical research, results with several significant ACF's and PACF's indicate a poor fit.

  • Residual analysis: A random normal distribution of residuals indicates a good ARIMA model (good identification of p, d, and 1). The Box-Ljung statistic should have significance levels of .05 or better for 95% of the time periods under analysis as a test of randomness. Box-Ljung statistics are created by SPSS under Analyze, Time Series, ARIMA; then click on the Display button and check autocorrelations and partial autocorrelations. P-P probability plot of residuals should form a 45-degree line as a test of normality. Failure to meet these tests suggests the need to transform data and/or to identify ARIMA parameters differently. In SPSS, P-P plots are found under Graphs, P-P, for plotting the variable ERR_1 (the residuals) created by Analyze, Time Series, ARIMA.
    • RMS (residual variance) is root mean square residual (called residual variance in SPSS and the variance estimate in SAS). It is an alternate means of assessing the residuals. As a goodness of fit measure, the smaller the RMS, the smaller the error and the better the fit of the model. That is, between two models, the one with the lower residual variance is the one with better fit. RMS is the mean square root of the squared residuals summed for all time periods.

      AIC is the Akaike Information Criterion and is a goodness of fit measure used to assess which of two ARIMA models are better, when both have acceptable residuals. The lower the AIC, the better the model. However, this comparison may only be made with nested models. (Ex., ARIMA (0,1,0) is nested under ARIMA(1,1,0); however, ARIMA (1,0,1) and ARIMA (0,1,0) cannot be compared by AIC because neither is nested under the other. There are also other goodness of fit measures less commonly used for this purpose, such as BIC (Bayesian Information Criterion) or SBC (Schwarz Bayesian Criterion).

  
  

• ARIMA Modeling: Intervention and Transfer Function Analysis
  • Adding interventions is simply using a dichotomous variable to divide the time periods into two sets - before an intervention event and after and intervention event - then seeing if the ARIMA (or other) models are the same. When the omega statistic is significant, the researcher concludes the intervention had a significant effect. Alternatively, the closer the delta statistic to 0, the more abrupt the effect of the intervention. SPSS produces omega; SAS produces both.
    Interpreting b coefficients of intervention variables. When an intervention variable coded 0= pre intervention, 1= post intervention, is added to the ARIMA model, whether the intervention significantly affects the time series variable (ex., whether the partial birth abortion ban significantly affected the number of abortions time series) can also be inferred from the significance of the intervention variable's b coefficient in the "Variables in the Equation" table of the ANOVA table in ARIMA output. If the b coefficient is not significant, the intervention was not significant. The value of b is the mean number of units in the dependent variable per time period (ex., mean number of abortions per year) change attributable to the intervention, controlling for autoregressive and moving average effects (if specified in the ARIMA model).

    In SPSS, one can also get case summary output using Analyze, Reports, Case Summaries. By doing this for cases before and after the intervention, SPSS will compute the median number of dependent variable units per time period before and after the intervention (ex., the median number of abortions per year).

    Control variables in intervention analysis. When additional independent variables beyond the intervention variable are added to the equation, these serve as controls. That is, the b coefficient of the intervention variable then reflects the intervention variable's effect on the dependent controlling for other variables in the equation, just as with ordinary regression, because the b coefficients are partial coefficients. Thus, for instance, if median income were added as a control in the example above, the value of b would be the mean number of abortions per year change attributable to the partial birth abortion ban, controlling for median income and controlling for autoregressive and moving average effects (if specified in the ARIMA model).

  • Adding transfer functions is adding continuous variables to the right-hand side of the time series equation, not including an intervention variable. As with ordinary regression (and as with adding interventions), the purpose of adding transfer functions is to see the effect on the dependent of additional independent variables other than previous values of the dependent itself. In the example, if median income is added but not a dichotomous variable for the partial birth abortion ban, then the b coefficient for median income would be the mean number of abortions per year change attributable to a unit change in median income, controlling for autoregressive and moving average effects (if specified in the ARIMA model).

  • A note on log transformed dependents. In both intervention and transfer function analysis, note that if the dependent variable is measured in log base 10 transform format, then the effect calculation is the antilog of 10 to the bth power to get mean number of units per time period associated with a unit change in the independent variable (which, in the case of intevention variables, is moving from 0=no intervention to 1=intervention).

  • Obtaining predictions. As with other forms of regression, ARIMA models can generate predicted values.
    • Setting the estimation and validation periods. The estimation (a.k.a. historical) period is the range of cases used to develop the ARIMA model. The validation period refers to hold-out time periods (rows in your dataset) used to test the model. In SPSS, this can be done outside ARIMA by selecting Data, Select cases, then click on the radio button for "Based on time or case range", then click the Range button and set the range for the estimation period. Then in SPSS, select Analyze, Time Series, ARIMA; then click the Save button and check the radio button you want:

    • Predict from estimation period through last case. This option is the default. It develops the ARIMA model based on the estimation time period, then predicts values for all cases from the estimation period through the end of the file (the last time period). If no estimation period was defined, all cases are used to predict values.

    • Predict through. Choose this option to predict into the future. This option develops the ARIMA model based on cases in the estimation period but the user can pick cases beyond the end of the dataset to predict for (ex., the data set ends in 2005 but the researcher can ask for predictions through 2050).

    • Output. With either choice above, SPSS will add additional columns to your dataset: FIT_1 is the prediction; ERR_1 is the residual (error) for the prediction; LCL_1 is the lower confidence limit on the estimate; UCL_1 is the upper confidence limit on the estimate; and SEP_1 is the standard error of fit.

Assumptions

1. Stationarity is a critical assumption of time series analysis, stipulating that statistical descriptors of the time series are invariant for different ranges of the series. Weak stationarity assumes only that the mean and variance are invariant. Strict stationarity also requires that the series is normally distributed. Stationarity is tested by the following tests: Durbin-Watson, Dickey-Fuller, Augmented D-F, and Root Examination for univariate time series. There is also a test (Fountis-Dickey) for joint stationarity when modeling two time series together. Stationarity may also be inspected graphically in SPSS by selecting Graphs, Sequence, to visually inspect for a linear or quadratic slope. Testing stationarity is a first step in time series modeling. These may be followed by tests for normality: the normal distribution test, Jarqua-Bear, or studentized range tests.

2. Uncontrolled autocorrelation. Time series analysis requires stationarity be established through differencing or some other technique. If two variables trend upward in raw data, as do GNP and entertainment expenditures, they will tend to correlate highly when a linear technique such as OLS (ordinary least-squares) regression is applied. In fact, many if not most nationally aggregated variables are of this type. For data in such series, the value of any given datum is largely determined by the value of the preceding datum in the series. This autocorrelation must be controlled before inferences may be made about correlation with other variables. Failure to control autocorrelation is vary apt to lead to spurious results, thinking there is a strong effect of, say, entertainment expenditures on GNP.

   More technically, significance tests of OLS regression estimates assume non-autocorrelation of the error terms. Error terms at sequential points in the series should constitute a random series. It is also assumed that the mean of the error terms will be zero (because estimates are half are above and half below the actual values), and the variance of the error terms will be constant throughout the time series. When, as in many time series, the value of a datum in time t largely determines the value of the subsequent datum in time t + 1, a dependency exists linking the error terms and the non-autocorrelation assumption is violated. The practical effect is that the significance of OLS estimates is computed to be far better than actual, leading the researcher to think that significant relationships exist when they do not. The Durbin-Watson test is the standard test for autocorrelation.

3. Applying Linear Techniques to Nonlinear Data. OLS regression assumes linear relationships. Applying linear techniques to nonlinear data will underestimate relationships and increase error of estimate. As with other uses of OLS regression, the linearity assumption is not violated by adding power or other nonlinear transform terms to the equation (ex., income-squared). The researcher must conduct tests for linearity. A common test is Ramsey's RESET test, discussed in the section on data assumptions. There are a variety of other tests for linear or nonlinear dependence, including the Keenan, Luukkonen, McLeod-Li, and Hsieh tests. If non-linearity is present, it may be possible to eliminate it by double differencing or data transformation (ex., logarithmic).

4. Arbitrary model lag order. Model lag order can have great effects on results. While tests exist to determine the optimal model order, these tests are purely statistical in nature. The researcher should have a theoretical basis establishing the face validity of the order of the model he or she has put forward.

5. No outliers. As in other forms of regression, outliers may affect conclusions strongly and misleadingly.

6. Random shocks. If shocks are present in the time series, they are assumed to be randomly distributed with a mean of 0 and a constant variance.

7. Uncorrelated random error. Residuals in a good time series model will be randomly distributed, exhibit a normal distribution, have non-significant autocorrelations and partial autocorrelations, and have a mean of 0 and homogeneity of variance over time. Correlated error does not bias estimates but does inflate standard errors, making statistical inference problematic. The Durbin-Watson test is the standard test for correlated error.

Frequently Asked Questions

• What are some common sources of time series data for social scientists?
  Many time series datasets are available through the Inter-University Consortium for Political and Social Research (ICPSR). Among the repeated surveys available are the General Social Survey (GSS) and the American National Election Studies (ANES), as wellas the Current Population Survey (CPS), the National Health Interview Survey (NHIS), and the Consumer Surveys from the Survey Research Center of the University of Michigan.

• What should I do about missing data?
  ARIMA can handle missing data in a series, though an empty observation must be supplied as a place-holder. Missing data are less important at the very beginning or end of a series, but when in the middle of a series, missing data will strongly affect commands such as Autoregression, Exponential Smoothing, Seasonal Decomposition, and Graphs/Spectral Plots. Before using any of these commands, the user must go to the SPSS Data Editor and replace the missing values, or the user can employ the Replace Missing Values command, which employes maximum likelihood estimation to fill in missing values.

• I suspect there is not a single trend line but rather the trend is different for different subgroups in my population. How do I handle this?
  While separate regressions might be run for each subgroup in the population, significance comparisons may become difficult because of different subgroup population sizes. It is therefore more advisable (and easier) to include the subgroups in the regression equation as dummy variables (ex., Democrat=1, non-Democrat=0) along with year as well as an interaction term (the dummy variable times year). The dependent variable will then equal a constant plus a regression coefficient times year, plus a regression coefficient times the dummy variable, plus a regression coefficient times the interaction term. If there is no group effect, the coefficients will be zero for the dummy variable and interaction term. Nonzero, same-sign coefficients for the dummy variable and interaction term would indicate that Democrats and non-Democrats are diverging on the dependent variable over time. Non-zero, different-sign coefficients would indicate that Democrats and non-Democrats are converging on the dependent variable over time, or the trend lines may even have crossed.

  Firebaugh (1997: 16) warns researchers to remember that one possible cause of group trends is differential recruitment into the groups rather than group effects per se. That is, Democrats and non-Democrats might be diverging on the abortion issue, for instance, because Democrats are attracting a larger percentage of pro-choice individuals to their ranks over time.

• A trend might be due to individuals changing their minds, or due to new individuals coming into my sample. How do I decompose the aggregate trend into individual and population turnover components?
  An aggregate change in the percent in favor of legalizing marijuana might be due to changed opinions or due to younger individuals coming into the sampled population, for instance. The easiest situation is where the aggregate change is due entirely to the individual component or entirely to the turnover component. The researcher keeps track of the pro-legalization percentage by cohort (range of years of birth of the respondents, such as born 1950-1960). If we have entirely an individual effect we would expect that the change within each cohort would approximate the aggregate change in the percent favoring legalization. If we have entirely a turnover effect we would expect that the percent favoring legalization would not change over time within any cohort, but new cohorts entering the sample would differ from existing ones.

  Often trends exhibit both individual and turnover effects. Firebaugh (1997: 22) recommends the construction of a cohort-by-period data array to assess relative effects. This is simply a table in which column 1 is the cohort ranges (ex., born 1950-1960, 1961 - 1970, etc). Subsequent columns are the percentages (ex., percentage favoring legalization of marijauna) for each of the repeated surveys (ex., 1985 survey, 1990 survey, etc.), The last columns are the percent changes between surveys (ex., percent change between 1985 and 1990, between 1990 and 1995, etc.). Such a data array allows the researcher to visually inspect changes by cohort by period using the same expectations discussed above.

  An alternative decomposition approach is to use regression, though this requires that the researcher to assume within-cohort changes are linear and additive. All respondents in all surveys are cumulated into a single dataset for this analysis. The regression formula sets the response of the a given respondent on a given year of survey equal to a constant plus a regression coefficient times year (of the survey for the given respondent) plus a regression coefficient times cohort (birth year of the given respondent). The regression coefficient for year is the within-cohort slope and the estimated effect of within-cohort change equals this coefficient times the difference in year of the last survey and year of the first (ex., 1995 - 1985 = 10). The regression coefficient for cohort is the cross-cohort slope and the estimated effect of cohort turnover equals this coefficient times the difference in average year of birth in the last survey minus the average year of birth in the first survey. The ratio of the two effects is the ratio of importance of individual versus turnover effects. The two effects will sum approximately (but not exactly) to the aggregate effect. A proof of this is given by Firebaugh (1997: 25-26), who provides more extensive discussion and additional strategies.

• How does one go about disentangling age, period, and cohort time series effects?
  This is a major focus of Firebaugh (1997). Age is the age of the individual surveyed. Period is the year of measurement. Cohort is the year of birth of the respondent. Therefore, cohort equals period minus age. Because of this equality, there is an identification problem: knowing any two fixes the value of the third. For instance, knowing period and age fixes the value of cohort. The researcher may be tempted to disentangle age, period, and cohort effects by including all three in the analysis, as by making all three independent variables in a regression equation. However, this is mathematically invalid because such a regression model is underidentified. This problem cannot be sidestepped by running three regressions, each with only two of these variables, because that will result in the same proportion of variance explained (R²) for each of the three models, because of the foregoing identity (that is, the fact that, for instance, age and period contain the same information as age and cohort).

  There is no solution to this identification problem, only strategies for trying to deal with it. Sometimes it is possible to fix the value of the regression coefficient for one of the effects, typically to zero. For instance, in a study of bisexual vs. homosexual preferences, one might assume that the age effect was zero and all changes over time were due to period and cohort effects. One could then use period and cohort as independents in a regression in which sexual preference was a dependent, but results would be invalid if the assumption that there was no age effect was an untrue assumption. A way of wrestling with such assumptions is to run three regressions, each time fixing one of the effects (age, cohort, period) to zero, then examining the resulting coefficients to assess whether, on the basis of external information, all three models seemed plausible. One may find, for instance, that a regression coefficient approaches zero for one of the models, yet one has reason to believe that the effect for that coefficient does indeed exist, meaning that that model is not plausible.

• Is there an acceptable ARIMA model for all data?
  No. Sometimes there is no specification of the p, d, and q parameters which will yield small random normal residuals devoid of autocorrelation.

Bibliography

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    The two most recent (1994) monographs in the QASS time series group survey univariate (QASS #99, by Cromwell, Labys, and Terraza,) and multivariate (QASS #100, by Cromwell, Hannan, Labys, and Terraza) tests for time series models. Univariate tests include those for stationarity to determine if differencing sequential time observations or some other process has eliminated general augmentation trends which may lead to spurious correlation and inference of causality. Other tests discussed test for normality, independence, linearity, heteroscedasticity, and autoregression, as well a procedures for testing model order and testing residuals. In the case of multivariate tests, tests cover joint stationarity, joint normality, independence, cointegration, causality, linear and nonlinear model specification, model order, and forecast accuracy. These two volumes are the most recent, and in some ways the most lucid, monographs in the set. Dozens of tests are included in the survey, and in each monograph an appendix comprehensively charts whether any of three computer packages (MicroTSP, RATS, and SHAZAM) implement the test, allow computation of the test, or if the test cannot be performed using the package. SAS and SPSS modules are not discussed.

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