This chapter’s objectives are to: Introduce the basic concept of cointegration and show that it applies in a variety of economic models, show that cointegration necessitates that the stochastic trends of nonstationary variables be linked | Chapter 6 Applied Economitric Time Series 4th ed. Walter Enders 1 Example of Cointegration and Money Demand In logarithms, an econometric specification for such an equation can be written as: mt = b0 + b1pt + b2yt + b3rt + et where: mt = demand for money pt = price level yt = real income rt = interest rate et = stationary disturbance term bi = parameters to be estimated 2 Other Examples Consumption function theory. Unbiased forward rate hypothesis. Commodity market arbitrage and purchasing power parity. The formal analysis begins by considering a set of economic variables in long-run equilibrium when b1x1t + b2x2t + + bnxnt = 0 Letting b and xt denote the vectors (b1, b2, , bn) and (x1t, x2t, , xnt), the system is in long-run equilibrium when bxt = 0. The deviation from long-run equilibrium—called the equilibrium error—is et, so that et = bxt 3 Generalization Letting β and xt denote the vectors (β1, β2, ., βn) and (x1t, x2t, ., xnt), the system is in long-run equilibrium when βxt = 0. The deviation from long-run equilibrium--called the equilibrium error--is et, so that: et = βxt If the equilibrium is meaningful, it must be the case that the equilibrium error process is stationary. 4 The scatter plot was drawn using the {y} and {z} sequences from Case 1 of Worksheet . Since both series decline over time, there appears to be a positive relationship between the two. The equilibrium regression line is shown. Three important points 1. Cointegration refers to a linear combination of non-stationary variables. If (β1, β2, . , βn) is a cointegrating vector, then for any non-zero value of λ, (λβ1, λβ2, . , λβn) is also a cointegrating vector. Typically, one of the variables is used to normalize the cointegrating vector by fixing its coefficient at unity. To normalize the cointegrating vector with respect to x1t, simply select λ = 1/β1. 2. The equation must be balanced in that the order of integration of the two sides must be equal 3. If xt has m . | Chapter 6 Applied Economitric Time Series 4th ed. Walter Enders 1 Example of Cointegration and Money Demand In logarithms, an econometric specification for such an equation can be written as: mt = b0 + b1pt + b2yt + b3rt + et where: mt = demand for money pt = price level yt = real income rt = interest rate et = stationary disturbance term bi = parameters to be estimated 2 Other Examples Consumption function theory. Unbiased forward rate hypothesis. Commodity market arbitrage and purchasing power parity. The formal analysis begins by considering a set of economic variables in long-run equilibrium when b1x1t + b2x2t + + bnxnt = 0 Letting b and xt denote the vectors (b1, b2, , bn) and (x1t, x2t, , xnt), the system is in long-run equilibrium when bxt = 0. The deviation from long-run equilibrium—called the equilibrium error—is et, so that et = bxt 3 Generalization Letting β and xt denote the vectors (β1, β2, ., βn) and (x1t, x2t, ., xnt), the system is in long-run equilibrium .
