TimeSeriesBook.pdf

# It corresponds to the identifying assumption with

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it corresponds to the identifying assumption with regard to B . Then, the effect starts to increase and reaches a maximum after three years and then declines monotonically. After 15 years the effect is practically zero. The 95- percent confidence intervals are rather large so that all effects are no longer statistically significant after a few number of years. 15.4.5 Example 2: IS-LM Model with Phillips Curve In this example we replicate the study of Blanchard (1989) which investigates the US business cycle within a traditional IS-LM model with Phillips curve. 18 The starting point of his analysis is the VAR(p) model: X t = Φ 1 X t - 1 + . . . + Φ p X t - p + C D t + Z t 18 The results do not match exactly those of Blanchard (1989), but are qualitatively similar.

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15.4. INTERPRETATION OF VAR MODELS 295 0 5 10 15 20 25 30 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 period effect of an advertisement shock on advertisement expenditures 0 5 10 15 20 25 30 -1 -0.5 0 0.5 1 1.5 2 2.5 effect of a sales shock on advertisement expenditures period 0 5 10 15 20 25 30 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 period effect of an advertisement shock on sales 0 5 10 15 20 25 30 -0.5 0 0.5 1 1.5 2 2.5 period effect of a sales shock on sales zero restriction
296 CHAPTER 15. INTERPRETATION OF VAR MODELS where { X t } is a five-dimensional time series X t = ( Y t , U t , P t , W t , M t ) 0 . The individual elements of X t denote the following variables: Y t . . . growth rate of real GDP U t . . . unemployment rate P t . . . inflation rate W t . . . growth rate of wages M t . . . growth rate of money stock . The VAR has attached to it a disturbance term Z t = ( Z yt , Z ut , Z pt , Z wt , Z mt ) 0 . Finally, { D t } denotes the deterministic variables of the model such as a constant, time trend or dummy variables. In the following, we assume that all variables are stationary. The business cycle is seen as the result of five structural shocks which impinge on the economy: V dt . . . aggregate demand shock V st . . . aggregate supply shock V pt . . . price shock V wt . . . wage shock V mt . . . money shock . We will use the IS-LM model to rationalize the restrictions so that we will be able to identify the structural form from the estimated VAR model. The disturbance of the structural and the reduced form models are related by the simultaneous equation system: AZ t = BV t where V t = ( V yt , V st , V pt , V wt , V mt ) 0 and where A and B are 5 × 5 matrices with ones on the diagonal. Blanchard (1989) proposes the following specification: (AD): Z yt = V dt + b 12 V st (OL): Z ut = - a 21 Z yt + V st (PS): Z pt = - a 34 Z wt - a 31 Z yt + b 32 V st + V pt (WS): Z wt = - a 43 Z pt - a 42 Z ut + b 42 V st + V wt (MR): Z mt = - a 51 Z yt - a 52 Z ut - a 53 Z pt - a 54 Z wt + V mt .

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15.4. INTERPRETATION OF VAR MODELS 297 In matrix notation the above simultaneous equation system becomes: 1 0 0 0 0 a 21 1 0 0 0 a 31 0 1 a 34 0 0 a 42 a 43 1 0 a 51 a 52 a 53 a 54 1 Z yt Z ut Z pt Z wt Z mt = 1 b 12 0 0 0 0 1 0 0 0 0 b 32 1 0 0 0 b 42 0 1 0 0 0 0 0 1 V dt V st V pt V wt V mt .
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• Spring '17
• Raffaelle Giacomini

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