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      Notice that rather than denoting the first constant with α, we chose to go with β₁. This is the more common convention in multivariate regression. To make the equation even more regular, we can assume that there is an X₁, which, unlike the other X's, is constant and always equal to one. This convention allows us to easily express a set of observations in matrix form. For t observations and n regressands, we could write:

      (3.93)

      where the first column of the X matrix —x₁₁, x₂₁, … , x_t1– is understood to consist entirely of ones. The entire equation can be written more succinctly as:

      (3.94)

      where, as before, we have used bold letters to denote matrices.

      MULTICOLLINEARITY

      In order to determine the parameters of the multivariate regression, we again turn to our OLS assumptions. In the multivariate case, the assumptions are the same as before, but with one addition. In the multivariate case, we require that all of the independent variables be linearly independent of each other. We say that the independent variables must lack multicollinearity:

      (A7) The independent variables have no multicollinearity.

      To say that the independent variables lack multicollinearity means that it is impossible to express one of the independent variables as a linear combination of the others.

      This additional assumption is required to remove ambiguity. To see why this is the case, imagine that we attempt a regression with two independent variables where the second independent variable, X₃, can be expressed as a linear function of the first independent variable, X₂:

(3.95)

If we substitute the second line of Equation 3.95 into the first, we get:

      (3.96)

      In the second line, we have simplified by introducing new constants and a new error term. We have replaced (β₁ + β₃λ₁) with β₄, replaced (β₂ + β₃λ₂) with β₅, and replaced (β₃ϵ₂ + ϵ₁) with ϵ₃. β₅ can be uniquely determined in a univariate regression, but there is an infinite number of combinations of β₂, β₃, and λ₂ that we could choose to equal β₅. If β₅ = 10, any of the following combinations would work:

      (3.97)

      This is why we say that β₂ and β₃ are ambiguous in the initial equation.

      Even in the presence of multicollinearity, the regression model still works in a sense. In the preceding example, even though β₂ and β₃ are ambiguous, any combination where (β₂ + β₃λ₂) equals β₅ will produce the same value of Y for a given set of X's. If our only objective is to predict Y, then the regression model still works. The problem is that the value of the parameters will be unstable. A slightly different data set can cause wild swings in the value of the parameter estimates, and may even flip the signs of the parameters. A variable that we expect to be positively correlated with the regressand may end up with a large negative beta. This makes interpreting the model difficult. Parameter instability is often a sign of multicollinearity.

      There is no well-accepted procedure for dealing with multicollinearity. The easiest course of action is often simply to eliminate a variable from the regression. While easy, this is hardly satisfactory.

      Another possibility is to transform the variables, to create uncorrelated variables out of linear combinations of the existing variables. In the previous example, even though X₃ is correlated with X₂, X₃ – λ₂X₂ is uncorrelated with X₂.

      (3.98)

      One potential problem with this approach is similar to what we saw with principal component analysis (which is really just another method for creating uncorrelated variables from linear combinations of correlated variables). If we are lucky, a linear combination of variables will have a simple economic interpretation. For example, if X₂ and X₃ are two equity indexes, then their difference might correspond to a familiar spread. Similarly, if the two variables are interest rates, their difference might bear some relation to the shape of the yield curve. Other linear combinations might be difficult to interpret, and if the relationship is not readily identifiable, then the relationship is more likely to be unstable or spurious.

      Global financial markets are becoming increasingly integrated. More now than ever before, multicollinearity is a problem that risk managers need to be aware of.

      Part VII Time Series Models

      Time series describe how random variables evolve over time and form the basis of many financial models.

      Random Walks

      A time series is an equation or set of equations describing how a random variable or variables evolves over time. Probably the most basic time series is the random walk. For a random variable X, with a realization x_t at time t, the following conditions describe a random walk:

(3.99)

      In other words, X is equal to its value from the previous period, plus a random disturbance, is mean zero, with a constant variance. The last assumption, combined with the fact that ϵt is mean zero, tells us that the ϵ's from different periods will be uncorrelated with each other. In time series analysis, we typically refer to x_t– 1 as the first lagged value of x_t, or just the first lag of x_t. By this convention, x_t– 2 would be the second lag, x_t– 3 the third, and so on.

      We can also think in terms of changes in X. Subtracting x_t– 1 from both sides of our initial equation:

      (3.100)

      In this basic random walk,

has all of the properties of our stochastic term, ϵt. Both are mean zero. Both have a constant variance, σ². Most importantly, the error terms are uncorrelated with each other. This system is not affected by its past. This is the defining feature of a random walk.

How does the system evolve over time? Note that Equation 3.99 is true for all time periods. All of the following equations are valid:

      (3.101)

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