Time Series Analysis

latent growth curve modeling

event history analysis

cox regression

Key Concepts and Terms

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 Concepts
- 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.
Exponential Smoothing
Regression Models (Curve-Fitting)

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.
- 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.
- 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.

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.

Assumptions

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.
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.
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).
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.
No outliers. As in other forms of regression, outliers may affect conclusions strongly and misleadingly.
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.
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?
What should I do about missing data?
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?
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?
How does one go about disentangling age, period, and cohort time-series effects?
Is there an acceptable ARIMA model for all data?
What are some common sources of time series data for social scientists?
What should I do about missing data?
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?
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?
How does one go about disentangling age, period, and cohort time series effects?
Is there an acceptable ARIMA model for all data?

Bibliography

Methodology

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@c 2006, 2008 G. David Garson
last update: 1/30/08.

Time Series Analysis

Key Concepts and Terms

Time series designs

Time Series Concepts

Exponential Smoothing

Regression Models (Curve-Fitting)

ARIMA Models

Assumptions

Frequently Asked Questions

Bibliography