The correlations discussed above are simple examples of autocorrelations (i.e. correlations involving a variable and a lag of itself). The autocorrelation function is a common tool used by researchers to understand the properties of a time series. Based on the material in the "Aside on lagged variables" and the "Aside on notation" from Chapter 8, we will use expressions like "the correlation between Y and lagged Y". We denote this as r₁.

In general, we may be interested in the correlation between Y and Y lagged p periods. For instance, our stock market data is observed monthly, so the correlation between Y and Y lagged p = 12 periods is the correlation between the stock price now and the stock price a year ago (i.e. a year is 12 months). We will denote this correlation by r_p and refer to it as "the autocorrelation at lag p". The autocorrelation function treats r_p as a function of p (i.e. it calculates r_p for p = 1, ..., P). P is the maximum lag length considered and is typically chosen to be quite large (e.g. P = 12 for monthly data). The autocorrelation function is one of the most commonly used tools in univariate time series analysis, precisely because it reveals quite a bit of information about the series.

Aside

1. r_p is the correlation between a variable (say, Y) and Y lagged p periods. In our discussion of r₁ we noted Y₁ lagged one period was Y₀, which did not exist. For this reason, we used data from t = 2, ..., T to define lagged Y and calculate r₁. An even more extreme form of the problem occurs in the calculation of r_p. Consider creating a new variable W which has observations W_t = Y_t for t = p + 1, ..., T and a new variable Z which has observations Z_t = Y_t−p for t = p + 1, ..., T. The correlation between W (i.e. Y) and Z (i.e. Y lagged p periods) is r_p. Note that each of the new variables contains T – p observations. So when we calculate r_p we are implicitly "throwing away" the first p observations. If we considered extremely long lags, we would be calculating autocorrelations with very few observations. In the extreme case, if we set p = T we have no observations left to use. This is a jus-tification for not letting p get too big. The issues raised in this paragraph are very similar to those raised in distributed lag models (see Chapter 8, "Aside on lagged variables").

2. The autocorrelation function involves autocorrelations with different lag lengths. In theory, we can use data from t = 2, ..., T to calculate r₁; data from t = 3, ..., T to calculate r₂; etc., ending with data from t = P + 1, ..., T to calculate r_p. But, note that this means that each autocorrelation is calculated with a different number of data points. For this reason, it is standard practice to select a maximum lag (P) and use data from t = P + 1, ..., T for calculating all of the autocorrelations.

Example: Stock prices on the NYSE (continued from page 139)

Table 9.1 presents the autocorrelation functions for Y = stock price index and ΔY = the return to the stock price index (using data from NYSE.XLS) using a maximum lag of 12 (i.e. P = 12). This information can also be presented graphically by making a bar chart with the lag length on the X-axis and the autocorrelation on the Y-axis, as in Figures 9.3 and 9.4.

A striking feature of Table 9.1 and Figures 9.3 and 9.4 is that autocorrelations tend to be virtually one for stock price variable even in the case of high lag lengths. In contrast, the autocorrelations for the change in the stock price are very small and exhibit a pattern that looks more or less random; the autocorrelations, in other words, are essentially zero. This pattern is common to many financial time series: the series itself has autocorrelations near one, but the change in the series has autocorrelations that are much smaller (often near zero). Below are a few ways of thinking about these autocorrelations:

Table 9.1. Autocorrelation functions.

Lag length (p)	Stock price	Change in stock price
1	0.9990	0.0438
2	0.9979	−0.0338
3	0.9969	0.0066
4	0.9958	0.0297
5	0.9947	0.0925
6	0.9934	−0.0627
7	0.9923	−0.0451
8	0.9912	−0.0625
9	0.9902	−0.0113
10	0.9893	−0.0187
11	0.9885	−0.0119
12	0.9876	0.0308

Figure 9.3. Autocorrelation function for stock prices.

1. Y is highly correlated over time. Even the stock price index a year ago (i.e. p = 12) is highly correlated with the stock price index today. ΔY does not exhibit this property. Stock returns this month are essentially uncorrelated with returns in previous months.

   

   Figure 9.4. Autocorrelation function for stock returns.

2. If you knew past values of the stock price index, you could make a very good estimate of what the stock price index was this month. However, knowing past values of stock returns will not help you predict the return this month.

3. Y "remembers the past" (i.e. it is highly correlated with past values of itself). This is an example of long memory behavior. ΔY does not have this property.

4. Y is a nonstationary series while ΔY is stationary. We have not formally defined the words "nonstationary" and "stationary", but they are quite important in time series econometrics. We will have more to say about them later, but note for now that the properties of the autocorrelation function for Y are characteristic of nonstationary series.

Exercise 9.3

Use the data on Y = the long-term interest rate given in INTERESTRATES.XLS.

1. Calculate the autocorrelation function for Y and ΔY with a maximum lag of 4 (i.e. P = 4).

2. Plot these autocorrelation functions in a bar chart.

3. Interpret the results you have obtained in (a) and (b).

The autocorrelation function is a useful tool for summarizing the properties of a time series. Yet, in Chapters 3 and 4, we argued that correlations have their limitations and that regression was therefore a preferable tool. The same reasoning holds here: autocorrelations, in other words, are just correlations, and for this reason it may be desirable to develop more sophisticated models to analyze the relationships between a variable and lags of itself. Many such models have been developed in the statistical literature on univariate time series analysis but the most common model, which can also be interpreted as a regression model, is the so-called autoregressive model. As the name suggests, it is a regression model where the explanatory variables are lags of the dependent variable (i.e. "auto" means "self" and hence an autoregression is a regression of a variable on lags of itself). The word "autoregressive" is usually shortened to "AR".

We begin by discussing the autoregressive model with the explanatory variable being the dependent variable lagged one period. This is called the AR(1) model:

for t = 2, ..., T. It looks exactly like the regression model discussed in previous chapters,^[49] except that the explanatory variable is Y_t−1. The value of φ in the AR(1) model is closely related to the behavior of the autocorrelation function and to the concept of nonstationarity.

In order to understand the types of behavior characteristic of the AR(1) series, let us artificially simulate three different time series using three different choices for φ : φ = 0, 0.8 and 1. All three series have the same values for a (i.e. α = 0.01) and the same errors. Figures 9.5, 9.6 and 9.7 provide time series plots of the three data sets.

Note that Figure 9.5 (with φ = 0) exhibits random-type fluctuations around an average of about 0.01 (the value of α). In fact, it is very similar to Figure 9.2, which contains a time series plot of stock returns. Figure 9.7 (with φ = 1) exhibits trend behavior and looks very similar to Figure 9.1, which plots the stock price level. Figure 9.6 (with φ = 0.8) exhibits behavior that is somewhere in-between the random fluctuations of Figure 9.5 and the strong trend of Figure 9.7.

Figures 9.5–9.7 illustrate the types of behavior that AR(1) models can capture and show why they are commonly used. For different values of φ, these models can allow for the randomly fluctuating behavior typical of growth rates of many financial time series; for the trend behavior typical of the financial series themselves; or for intermediate cases between these extremes.

Note also that φ = 1 implies the type of trend behavior we have referred to as non-stationary above, while the other values of φ imply stationary behavior. This allows us to provide a formal definition of the concepts of stationarity and nonstationarity, at least for the AR(1) model: For the AR(1) model, we can say that Y is stationary if |φ| < 1 and is nonstationary if φ = 1. The other possibility, |φ| > 1, is rarely considered in finance. The latter possibility implies that the time series is exhibiting explosive behavior over time. Since such explosive behavior is only observed in unusual cases, it is of little empirical relevance and we shall not discuss it here. Mathematical intuition for the properties of the AR(1) model and how it relates to the issue of nonstationarity is given in Appendix 9.1.

Figure 9.5. AR(1) time series with φ = 0.

Figure 9.6. AR(1) time series with φ = 0.8.

Figure 9.7. AR(1) time series with φ = 1.

Above we introduced the terms "nonstationary" and "stationary" without providing any formal definition (except for the AR(1) model). As we shall see, the distinction between stationary and nonstationary time series is an extremely important one. To formally define these concepts requires that we get into statistical issues that are beyond the scope of this book. But we provide some general intuition for these concepts below.

Formally, "nonstationary" merely means "anything that is not stationary". Economists usually focus on the one particular type of nonstationarity that seems to be present in many financial time series: unit root nonstationarity. We will generalize this concept later, but at this stage it is useful to think of a unit root as implying φ = 1 in the AR(1) model. Following are different ways of thinking about whether a time series variable, Y, is stationary or has a unit root:

The final point can be seen most clearly by subtracting Y_t−1 from both sides of the equation in the AR(1) model, yielding:

where ρ = φ – 1. Note that, if φ = 1, then ρ = 0 and the previous equation can be written solely in terms of ΔY_t, implying that ΔY_t fluctuates randomly around α. For future reference, note that we can test for ρ = 0 to see if a series has a unit root. Furthermore, a time series will be stationary if −1 < φ < 1 which is equivalent to −2 < ρ < 0. We will refer to this as the stationarity condition.

By way of providing more intuition (and jargon!) for the AR(1) model, let us consider the case where φ = 1 (or, equivalently, ρ = 0). In this case we can write the AR(1) model as:

This is referred to as the random walk model. More precisely, the random walk model has no intercept (i.e. α = 0), while the preceding equation is referred to as a random walk with drift. The presence of the intercept allows for changes in variables to be, on average, non-zero. So, for instance, if Y is (the log of) a stock price of a particular company then the random walk with drift model can be written as:

Since the left-hand of this equation is the stock return (exclusive of dividends), then the model says stock returns are equal to α (e.g. a benchmark return relevant for this company taking into account its risk, etc.) plus a random error.

In the random walk model, since φ = 1, Y has a unit root and is nonstationary. This model is commonly thought to hold for phenomena like stock prices, a point we will elaborate on next.

Example: Stock prices on the NYSE (continued from page 143)

The AR(1) model is a regression model. Accordingly, we can use OLS to regress the variable Y on lagged Y.^[50] If we do this, we find

If we regress ΔY_t on Y_t−1, we obtain an OLS estimate of ρ which is −0.00014. Note that we are finding

We have argued above that the AR(1) model can be interpreted as a simple regression model where last period's Y is the explanatory variable. However, it is possible that more lags of Y should be included as explanatory variables. This can be done by extending the AR(1) model to the autoregressive of order p, AR(p), model:

for t = p + 1, ..., T. We will not discuss the properties of this model, other than to note that they are similar to the AR(1) model but are more general in nature. That is, this model can generate the trend behavior typical of financial time series and the randomly fluctuating behavior typical of their growth rates.

In discussing unit root behavior it is convenient to subtract Y_t−1 from both sides of the previous equation. With some rearranging^[51] we obtain:

where the coefficients in this regression, ρ, γ₁, ..., γ_p−1 are simple functions of φ₁,..., φ_p. For instance, ρ = φ₁ + ...+ φ_p – 1. Note that this is identical to the AR(p) model, but is just written differently. Hence we refer to both previous equations as AR(p) models. In case you are wondering where the Y_t−p term from the first equation went to in the second, note that it appears in the second equation in the ΔY_t−p+1 term (i.e. ΔY_t−p+1 = Y_t−p+1 – Y_t−p). Note also that both variants have the same number of coefficients, p + 1 (i.e. the first variant has α, φ₁, ..., φ_p while the second variant has α, ρ, γ₁, ..., γ_p−1). However, in the second variant the AR(p) model has last coefficient γ_p−1. Don't let this confuse you, it is just a consequence of the way we have rearranged the coefficients in the original specification.

The key points to note here are that the above equation is still in the form of a regression model; and ρ = 0 implies that the AR(p) time series Y contains a unit root; if −2 < ρ < 0, then the series is stationary. Looking at the previous equation with ρ = 0 clarifies an important way of thinking about unit root series which we have highlighted previously: if a time series contains a unit root then a regression model involving only ΔY is appropriate (i.e. if ρ = 0 then the term Y_t−1 will drop out of the equation and only terms involving ΔY or its lags appear in the regression). It is common jargon to say that "if a unit root is present, then the data can be differenced to induce stationarity".

As we will discuss in the next chapter, with the exception of a case called cointegration, we do not want to include unit root variables in regression models. This suggests that, if a unit root in Y is present, we will want to difference it and use ΔY. In order to do so, we must know first if Y has a unit root. In the past, we have emphasized that unit root series exhibit trend behavior. Does this mean that we can simply examine time series plots of Y for such trending to determine if it indeed has a unit root? The answer is no. To explain why, let us introduce another model.

We showed previously that many financial time series contain trends and that AR models with unit roots also imply trend behavior. However, there are other models that also imply trend behavior. Imagine that Figure 9.1 (or Figure 9.7) is an XY-plot where the X-axis is labeled time, and that we want to build a regression model using this data. You might be tempted to fit the following regression line:

where the coefficient on the explanatory variable, time, is labeled δ to distinguish it from the φ in the AR(1) model. Note that you can interpret the previous regression as involving the variable Y and another variable with observations 1, 2, 3, 4, ..., T. This is another regression model which yields trend behavior. To introduce some jargon, the term δt is referred to as a deterministic trend since it is an exact (i.e. deterministic) function of time. In contrast, unit root series contain a so-called stochastic trend (justification for the term "stochastic trend" is given in Appendix 9.1).

We can even combine this model with the AR(1) model to obtain:

Figure 9.8 is a time series plot of artificial data generated from the previous model with α = 0, φ = 0.2 and δ = 0.01. Note that this series is stationary since |φ| < 1. Yet, Figure 9.8 looks much like Figure 9.7 (or Figure 9.1). Stationary models with a deterministic trend can yield time series plots that closely resemble those from non-stationary models having a stochastic trend. Thus, you should remember that looking at time series plots alone is not enough to tell whether a series has a unit root.

Figure 9.8. Trend stationary series.

The discussion in the previous paragraph motivates jargon that we will use and introduce in the context of the following summary:

If we add a deterministic trend to the AR(p) model, we obtain a very general model that is commonly used in univariate time series analysis:

We refer to the above as the AR(p) with deterministic trend model and use it later. You may wonder why we don't just use the original AR(p) specification introduced at the beginning of this section (i.e. the one where the explanatory variables are Y_t-1, ..., Y_t−p). There are two reasons. First, we are going to test for a unit root. With the present specification, this is simply a test of ρ = 0. Testing for whether regression coefficients are zero is a topic which we have learned previously (refer to Chapter 5). With the original AR(p) model, testing for a unit root is more complicated. Second, Y_t−1, Y_t−2, ..., Y_t−p are often highly correlated with each other (see the autocorrelation function in Figure 9.3). If we were to use them as explanatory variables in our regression we might run into serious multicollinearity problems (see Chapter 6). However, in the present model we use Y_t−1, ΔY_t−1, ..., ΔY_t−p+1 as explanatory variables, which tend not to be highly correlated (see Figure 9.4), thereby avoiding the problem.

Example: Stock prices on the NYSE (continued from page 148)

The following table contains output from an OLS regression of ΔY_t on Y_t−1, ΔY_t−1, ΔY_t−2, ΔY_t−3 and a deterministic time trend, created by using the data on stock prices from NYSE.XLS. In other words, it provides regression output for the AR(4) with deterministic trend model. We suspect that stock prices may contain a unit root, a supposition supported somewhat by the table. In particular, a unit root is present if ρ (the coefficient on Y_t−1) is zero. As we can see, the estimate of ρ is indeed very small (i.e.

In Chapters 5 and 6, we described how to test whether regression coefficients were equal to zero. These techniques can be used in the AR(p) with deterministic trend model (i.e. if you wish to omit explanatory variables whose coefficients are not significantly different from zero). In particular, testing is usually done to help choose lag length (p) and to decide whether the series has a unit root. In fact, it is common to first test to select lag length (using hypothesis tests as described in Chapter 8), and then test for a unit root.

However, there is one important complication that occurs in the AR(p) model that was not present in earlier chapters. To understand it, let us divide the coefficients in the model into two groups: (1) α, γ₁, ..., γ_p−1, and Δ, and (2) ρ. In other words, we consider hypothesis tests involving ρ separately from those involving the other coefficients.

Testing involving α, γ₁, ..., γ_p−1, and δ

Many sophisticated statistical criteria and testing methods exist to determine the appropriate lag length in an AR(p) model. Nonetheless, simply looking at the t-statistics or P-values in regression outputs can be quite informative. For instance, an examination of Table 9.2 reveals that the P-values associated with the coefficients on the lagged ΔY terms are insignificant, and that they may be deleted from the regression (i.e. the P-values are greater than 0.05). Alternatively, a more common route is to proceed sequentially, as we did in Chapter 8; that is, to choose a maximum lag length, p_max, and then sequentially drop lag lengths if the relevant coefficients are insignificant.

More specifically, begin with an AR(p_max). If the p_maxth lag is insignificant, we reduce the model to an AR(p_max – 1). If the (p_max – 1)th lag is insignificant in the AR(p_max – 1) then drop it and use an AR(p_max – 2), etc. Generally, you should start with a fairly large choice for p_max.

In the AR(p) with deterministic trend model we also have to worry about testing whether δ = 0. This can be accomplished in the standard way by checking whether its P-value is less than the level of significance (e.g. 0.05). This test can be done at any stage, but it is common to carry it out after following the sequential procedure for choosing p.

A short summary of this testing strategy is outlined below:

Step 1. Choose the maximum lag length, p_max, that seems reasonable to you.

Step 2. Estimate using OLS the AR(p_max) with deterministic trend model:

If the P-value for testing γp_max−1 = 0 is less than the significance level you choose (e.g. 0.05) then go to Step 5, using p_max as lag length. Otherwise go on to the next step.

Step 3. Estimate the AR(p_max – 1) model:

If the P-value for testing γp_max−2 = 0 is less than the significance level you choose (e.g. 0.05) then go to Step 5, using p_max – 1 as lag length. Otherwise go on to the next step.

Step 4. Repeatedly estimate lower order AR models until you find an AR(p) model where γ_p−1 is statistically significant (or you run out of lags).

Step 5. Now test for whether the deterministic trend should be omitted; that is, if the P-value for testing δ = 0 is greater than the significance level you choose then drop the deterministic trend variable.

Example: Stock prices on the NYSE (continued from page 152)

If we carry out the preceding strategy on the NYSE stock price data, beginning with p_max = 4, the model reduces to:

That is, we first estimated an AR(4) with deterministic trend (see Table 9.2) and found the coefficient on ΔY_t−3 to be insignificant. Accordingly, we estimated an AR(3) with deterministic trend and found the coefficient on ΔY_t−2 to be insignificant. We then dropped the latter variable and ran an AR(2), etc. Eventually, after finding the deterministic trend to be insignificant, we settled on the AR(1) model. OLS estimation results for this model are given in Table 9.3.

Table 9.3. AR(1) model.

	Coefficient	Standard error	t-stat	P-value	Lower 95%	Upper 95%
Intercept	0.00763	0.0123	0.6188	0.5363	−0.0166	0.0318
Yt−1	−0.00012	0.0019	−0.0631	0.9497	−0.0039	−0.0037

Table 9.3 shows the table of results you might report in a paper or empirical project, including a brief but coherent explanation of the strategy that you used to arrive at this final specification.

These results lead us to the next, most important, testing question: does Y contain a unit root? Remember that, if ρ = 0, then Y contains a unit root. In this case, the series must be differenced in the regression model (i.e. it is difference stationary). You may think that you can simply test ρ = 0 in the same manner as you tested the significance of the other coefficients. For instance, you might think that by comparing the P-value to a significance level (e.g. 0.05), you can test for whether ρ = 0. SUCH A STRATEGY IS INCORRECT! In hypothesis testing, ρ is different from other coefficients and, thus, we must treat it differently.

Testing involving ρ

To fully understand why you cannot carry out a unit root test of ρ = 0 in the same manner as we would test other coefficients requires that you have knowledge of statistics beyond that covered in this book. Suffice it to note here that most regression packages like Excel implicitly assume that all of the variables in the model are stationary when they calculate P-values. If the explanatory variable Y_t−1 is nonstation-ary, its P-value will be incorrect. A correct way of testing for a unit root has been developed by two statisticians named Dickey and Fuller and is known as the Dickey—Fuller test.^[52] They recommend using the t-statistic for testing ρ = 0, but correcting the P-value.

We can motivate the Dickey–Fuller test in terms of the following: in Chapter 5, we said that testing could be done by comparing a test statistic (here the t-stat) to a critical value to determine whether the former was either "small" (in which case the hypothesis was accepted) or "large" (in which case the hypothesis was rejected). In the standard (stationary) case, the critical values are taken from statistical tables of the Student-t distribution. Dickey and Fuller demonstrated that in the unit root case, this is incorrect. They calculated the correct statistical tables from which to take critical values.

The previous paragraphs were meant to motivate why the standard testing procedure was incorrect. Admittedly, they are not very helpful in telling you what to do in practice. If you are going to work extensively with time series data, it is worthwhile to either:

1. Use a computer software package that is more suitable for time series analysis than Excel. Packages such as Stata, E-views, MicroFit or SHAZAM will automatically provide you with correct critical values or P-values for your unit root test. As before, you will reject the unit root if the P-value is less than 0.05 or if the t-stat is greater than the critical value (in an absolute value sense).

2. Read further in time series econometrics and learn how to use the Dickey–Fuller statistical tables.^[53]

However, a rough rule of a thumb can be used that will not lead you too far wrong if your number of observations is moderately large (e.g. T > 50). This approximate rule is given in the following strategy for testing for a unit root:

1. Use the strategy outlined in Steps 1 to 5 above to estimate the AR(p) with deterministic trend model. Record the t-stat corresponding to ρ (i.e. the t-stat for the coefficient on Y_t−1).

2. If the final version of your model includes a deterministic trend, the Dickey–Fuller critical value is approximately −3.45. If the t-stat on ρ is more negative than −3.45, reject the unit root hypothesis and conclude that the series is stationary. Otherwise, conclude that the series has a unit root.

3. If the final version of your model does not include a deterministic trend, the Dickey–Fuller critical value is approximately −2.89. If the t-stat on ρ is more negative than −2.89, reject the unit root hypothesis and conclude that the series is stationary. Otherwise, conclude that the series has a unit root.^[54]

In the previous example, the final version of the AR(p) model did not include a deterministic trend. The t-stat on ρ is −0.063, which is not more negative than −2.89. Hence we can accept the hypothesis that NYSE stock prices contain a unit root and are, in fact, a random walk. Be careful using this crude rule of thumb when your t-stat is close to the critical values listed here.

Example: Long-term interest rates

If we carry out the preceding strategy on the long-term interest rate data (from INTERESTRATES.XLS), beginning with p_max = 4 and sequentially deleting insignificant lagged variables, we end up with an AR(1) model:

OLS estimation results for this model are given in Table 9.4.

We are interested in testing for a unit root and this occurs if ρ = 0. A naïve researcher who did not know about the Dickey–Fuller test would incorrectly say: "Since the P-value for ρ (which is 0.035) is less than 0.05, we can conclude that ρ is significant. Thus, the long-term interest rate variable does not contain a unit root". However, a researcher who knew about the Dickey–Fuller test would say: "The final version of the AR(p) model I used did not include a deterministic trend. Hence, I must use the Dickey–Fuller critical value of −2.89. The t-stat on ρ is −2.13, which is not more negative than −2.89. Hence we can accept the hypothesis that long-term interest rates contain a unit root".

Table 9.4. AR(1) model.

	Coefficient	Standard error	t-stat	P-value	Lower 95%	Upper 95%
Intercept	0.039	0.014	2.682	0.008	0.010	0.067
Yt−1	−0.004	0.002	−2.130	0.035	−0.0077	−0.0003

Exercise 9.7

In this chapter we have recommended a strategy according to which you begin with an AR(p) with deterministic trend model, choose lag length (p), decide whether the deterministic trend should be included or excluded, and then test for a unit root. Carry out this strategy using the following series:

1. Those in FIG95.XLS and FIG96.XLS (which you know are stationary).

2. That in FIG97.XLS (which you know has a unit root).

3. That in FIG98.XLS (which is trend stationary, but exhibits strong trending behavior).

4. The short term interest rate variable in INTERESTRATES.XLS.

Exercise 9.8

In Exercise 9.7, we tested for unit roots in many series. We noted that if a time series has one unit root, then its difference will be stationary. Verify that this is true for the series having unit roots in Exercise 9.7. That is, discuss how you would test to see if the change in the series has a unit root. Then carry out this test.

Some words of warning about unit root testing: The Dickey–Fuller test exhibits what statisticians refer to as low power. In other words, the test can make the mistake of finding a unit root even when none exists. Intuitively, trend stationary series can look a lot like unit root series (compare Figures 9.7 and 9.8) and it can be quite hard to tell them apart. Furthermore, other kinds of time series models can also appear to exhibit unit root behavior, when in actuality they do not have unit roots. A prime example is the time series model characterized by abrupt changes or breaks. These structural breaks can occur in time series models, and can be precipitated by events such as wars or crises in supply (e.g. the OPEC oil embargo). Stock prices can exhibit structural breaks due to market crashes; and commodity prices, due to droughts and other natural disasters. All in all, structural breaks are potentially a worry for many types of time series data and some caution needs to be taken when interpreting the results of Dickey–Fuller tests. There are many other tests for a unit root. To explain these would require a long technical digression which would distract from the main focus of this book. However, depending on which software package you have available, a little bit of extra study should enable you to use them in practice. Remember the structure of any hypothesis testing procedure involves your knowing the hypothesis being tested (here the unit root hypothesis) as well as a P-value (or a test statistic with critical value to compare it to). Many software packages provide these automatically. For instance, Stata allows you to do another popular unit root test called the Phillips–Perron test (as well as others) in this manner.

Exercise 9.9

This exercise could actually be interpreted as an empirical project and could be short or long, depending on what the student is interested in achieving. The book Nonlinear Time Series Models in Empirical Finance by Philip Hans Franses and Dick van Dijk (Cambridge University Press) is one you may use in your future studies. It uses more sophisticated tools in statistics than the present book, so at this stage you might find it difficult. However, it does have a website containing a rich collection of data sets from several countries on stock prices and exchange rates which you can download (see http://www.few.eur.nl/few/people/djvandijk/nltsmef/nltsmef.htm). In particular, stock price indices from Amsterdam (EOE), Frankfurt (DAX), Hong Kong (Hang Seng), London (FTSE100), New York (S&P500), Paris (CAC40), Singapore (Singapore All Shares) and Tokyo (Nikkei) are provided. The exchange rates are the Australian dollar, British pound, Canadian dollar, German DeutschMark, Dutch guilder, French franc, Japanese yen and the Swiss franc, all expressed as number of units of the foreign currency per US dollar. The sample period for the stock indexes runs from 6 January 1986 until 31 December 1997, whereas for the exchange rates the sample covers the period from 2 January 1980 until 31 December 1997. Investigate the random walk hypothesis using this data. Do stock prices appear to follow a random walk in every country? Do exchange rates?

Note that this data is available at a daily frequency. You may want to work with the data at this frequency or at a weekly (e.g. by just using data every Wednesday) or monthly frequency (e.g. by just using data from the last day of each month). Do your results for the unit root tests depend on whether you use daily, weekly or monthly data?

Mathematical insight into the properties of the AR(1) model can be gained by writing it in a different way. For simplicity, we will set α = 0 in order to focus on the role that lagged Y plays. Note that the AR(1) model will hold at any point in time so we can lag the whole AR(1) equation given in the body of the chapter and write:

If we substitute this expression for Y_t−1 into the original AR(1) model we obtain:

Note that the previous expression depends on Y_t−2, but we can write:

and substitute this expression for Y_t−2 in the other equation. If this procedure is repeated we end up with an alternative expression for the AR(1) model:

This expression looks complicated, but we can consider two special cases as a means of understanding its implications. In the first of these we assume φ = 1 and the previous equation reduces to:

The important point to note about the two terms on the right hand side of the previous equation is that they illustrate a long memory property; the value of the time series starts at Y₁, which always enters the expression for Y_t, even if t becomes very large. That is, the time series "never forgets" where it started from. It also "never forgets" past errors (e.g. e₁ always enters the above formula for Y_t even if t gets very large). It can be shown that the trending behavior of this model arises as a result of the second term, which says that current Y contains the sum of all past errors. Statisticians view these errors as random or "stochastic" and this model is often referred to as containing a stochastic trend. This is a key property of nonstationary series.

A second special case stands in contrast to the properties described above. If we suppose |φ| < 1, we can see that φ^t−1 will be decreasing as t increases (e.g. if φ = 0.5, then φ² = 0.25, φ¹⁰ = 0.001 and φ¹⁰⁰ = 7.89 × 10⁻³¹, etc.). The influence of Y₁ and past errors on Y_t will gradually lessen as t increases and Y slowly "forgets the past". Y will not exhibit the long memory property we observed for the case where φ = 1. This is a key property of stationary series.