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# Trends

BIBLIOGRAPHY

A trend is a relatively smooth and unidirectional pattern in data that arises from the accumulation of information over time. Many of the time series in economics and in other social sciences exhibit a smooth (upward or downward) tendency. In developed economies, the macro-socioeco-nomic time series, such as gross national product, consumption, income, and population, are characterized by an upward trend. Though in the short run the economy goes through expansions and recessions, in the long run there is growth. This is due to technological progress and/or population growth. The trend arises from the accumulation of socioeconomic activity from one period to the next. For instance, the time series of the U.S. population has an upward trend over time because the number of births exceeds the number of deaths and/or because the number of immigrants exceeds the number of emigrants.

From an econometric point of view there is a distinction between deterministic and stochastic trends, depending on whether the accumulation of information is due to a deterministic or to a random component. Suppose that an economic variable Yt, for instance national product, can be represented by a model such as Yt = a + bt + εt for t = 1,2,3,. In this case, the trend is a line a + bt with a as intercept and b as slope, which is also the growth rate. Because t = 1,2,3, is purely deterministic, the trend is deterministic. An example of an exponential deterministic trend is the time series of the U.S. population. Conversely, a stochastic trend is generated by the accumulation of random components. An example of a model with a stochastic trend is Yt = εt + εt -1 + εt -2 + ε 1 where {εt} is a collection of normal random variables. An example of a stochastic trend is the time series of speculative prices such as stock prices or exchanges rates. The time series generated by deterministic trend models tend to be smoother than those generated by a stochastic trend.

Deterministic trends may have different shapes. Among these, the most common are linear, quadratic, exponential/logarithmic, and logistic. Only when the trend is linear is the growth rate constant. In any other specification, the growth rate will depend on the time period under study. Models with a deterministic trend are also known as trend-stationary models because all the unconditional moments, with the exception of the mean, are time-invariant. Their estimation and testing is straightforward because standard statistical results apply. In the linear trend model specified above, the parameters a and b can be estimated by Ordinary Least Squares (OLS), assuming that the error term is well-behaved. Hypothesis testing relies on tests that have standard asymptotic distributions, such as the t-ratio and F-tests.

Stochastic trends are more complicated than deterministic trends because they are non-stationary. Their statistical analysis requires the use of non-standard asymptotic distributions, which need to be obtained by simulation techniques. The model introduced above, Yt = εt + εt -1 + εt -2 + ε 1, has the following equivalent representation Yt = Yt-1 + εt, which is an autoregressive process of order one with an autoregressive parameter equal to one. This model is also known as a unit root process or random walk without drift. If the model includes a constant such as Yt = c + Y t -1 + εt, one says that the process is a random walk with drift . The drift produces a smoother upward (or downward) tendency. Unit root processes are non-stationary because the unconditional moments (mean, variance, and covariances) are increasing functions of time. The autocorrelation function of a unit root is very characteristic with autocorrelation coefficients of any order asymptotically equal to one.

From an empirical perspective it is important to differentiate a deterministic trend model from a stochastic trend model. There are two main reasons for this: the need to conduct the correct statistical inference, and the construction and interpretation of the correct forecast. Consequently, the first step in empirical research is testing for unit root, which is a test of non-stationarity versus sta-tionarity. In a model such as Yt = c + φYt 1 + εt, the null hypothesis is set as H 0:φ = 1 (non-stationarity) versus an alternative H 1: φ < 1 (stationarity). Under the null, the standard t-ratio is not normally distributed but has a nonstandard distribution, which is known as the Dickey-Fuller (DF) distribution. The DF critical values are tabulated by numerical simulation. A rejection of the null in favor of the alternative hypothesis means that the process does not have a stochastic trend. On the contrary, failure to reject the null hypothesis means that there is not enough evidence against the unit root process and a stochastic trend in the data should be entertained.

From a forecasting perspective, a deterministic trend model produces a forecast along the time trend specification. The uncertainty associated with the forecast is bounded regardless of how far into the future one wishes to predict. However, a forecast from a stochastic trend model has unbounded uncertainty as the variance of the forecast is an increasing function of time.

A pervasive problem in empirical research in the social sciences during the 1970s and 1980s is the case of spurious regression. Regression is spurious when two (or more) variables, Yt and Xt, are found to be correlated but in fact they are not. This finding arises because Yt and Xt are non-stationary (unit root processes) and a regression of Yt on Xt does not take into account the stochastic trend. The diagnosis of spurious regression is relatively simple. If the R-squared of the regression is extremely high, around 0.90 and above, the t-statistics are exceptionally large, and the Durbin-Watson statistic is low, it is highly likely that the regression of Yt on Xt is spurious and, consequently, there is no correlation between them. The correct approach to analyze the correlation between Yt and Xt is first to remove the stochastic trend and then to run the regression. By first-differencing the data, the stochastic trend is removed. For instance, if Yt and Xt have a unit root, for example Yt = Yt-1 + εt and Xt = Xt-1 + Vt, the first difference of Yt is Δ Yt = εt and the first difference of Xt is ΔXt = νt. The proper regression is to regress ΔYt on ΔXt. However, if Yt and Xt have a deterministic trend, the data should not be first-differenced but the regression of Yt on Xt should contain a deterministic trend specification. There is only one instance in which a regression of Yt on Xt, being both unit root processes, is meaningful. This is the case of cointegration, in which Yt and Xt share the same stochastic trend. In this instance there is a longrun equilibrium between both processes.

SEE ALSO Autoregressive Models; Least Squares, Ordinary; Random Walk; Regression Analysis; Time Series Regression; Unit Root and Cointegration Regression

## BIBLIOGRAPHY

Dickey, David A., and Wayne A. Fuller. 1979. Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association 74: 427431.

Hamilton, James D. 1994. Time Series Analysis. Princeton, NJ: Princeton University Press.

Gloria González-Rivera