# Power 2

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## Power 2

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1. Power 2 Econ 240C

2. Lab 2 Retrospective • Exercise: • GDP_CAN = a +b*GDP_CAN(-1) + e • GDP_FRA = a +b*GDP_FRA(-1) + e

3. Data in Excel

5. Data in Excel

6. Stacking • So the dependent variable starts with gdp_can(1951) and goes through gdp_can(1992). Then the next value in the stack is gdp_fra(1951) and the data continues ending with gdp_fra(1992). • The independent variable for Canada starts with gdp_can(1950) and goes through gdp_can(1991). Then the rest of the stack is 42 zeros

7. Stacking • The independent variable for France starts with a stack of 42 zeros. Then the next observation is gdp_fra(1950), the following is gdp_fra(1951) etc. ending with gdp_fra(1991) • The constant stack for Canada is 42 ones followed by 42 zeros • The constant term for France is 42 zeros followed by 42 ones

8. Outline • Time Series Concepts • Inertial models • Conceptual time series components • Simulation and synthesis • Simulated white noise, wn(t) • Spreadsheet, trace, and histogram of wn(t) • Independence of wn(t)

9. Univariate Time Series Concepts • Inertial models: Predicting the future from own past behavior • Example: trend models • Other example: autoregressive moving average (ARMA) models • Assumption: underlying structure and forces have not changed

10. Conceptual time series components model • Time series = trend + seasonal + cycle + random • Example: linear trend model • Y(t) = a + b*t + e(t) • Example linear trend with seasonal • Y(t) = a + b*t + c1*Q1(t) + c2*Q2(t) + c3*Q3(t) + e(t)

11. How to model the cycle? • We have learned how to model: • Trend: linear and exponential • Seasonality: dummy variables • Error: e.g. autoregressive • How do you model the cyclical component?

12. Cyclical time series behavior • Many economic time series vary with the business cycle • Model the cycle using ARMA models • That is what the first half of 240C is all about

13. Simulation and Synthesis • Build ARMA models from noise, white noise, in a process called synthesis • The idea is to start with a time series of simple structure, and build ARMA models by transforming white noise

14. Simulated white noise • Generate a sequence of values drawn from a normal distribution with mean zero and variance one, i.e. N(0, 1) • In EViews: Gen wn = nrnd

15. The first ten values of simulated white noise, wn(t) Valuedraw = time index -0.628093683959 1 -0.627803051549 2 0.00723255412415 3 1.94192735344 4 -1.10119663665 5 0.514236967572 6 -0.843129585702 7 -0.0153352207678 8 1.25353192311 9 1.48589824393 10

16. Trace (plot) of first 100 values of wn(t) No obvious time Dependence, i.e. Stationary, not Trended, not seasonal

17. Histogram and Statistics, 1000 Obs.

18. Independence • We know each drawn value is from the same distribution, i.e. N(0,1) • We know every value drawn is independent from all other values • So wn(t) should be iid, independent and identically distributed

19. Independence: conceptual • Suppose the mean series, m(t), of white noise is zero, i.e. E wn(t) = m(t) = 0 • This is a good suppose because every generated value has expectation zero since it is from N(0,1) • Then E[wn(t)*wn(t-1)] = 0, i.e. a value is independent from the previous or lagged value

20. Independence: conceptual • In general: cov [wn(t)*wn(t-u)], where wn(t-u) is lagged u periods from t is defined as cov[wn(t)*wn(t-u)] = E{[wn(t) – Ewn(t)]*[wn(t-u) – Ewn(t-u)]} = E [wn(t)*wn(t-u)], since E wn(t) = 0 • This is called the autocovarince, i.e. the covariance of white noise with lagged values of itself

21. Independence: Conceptual • For every value of lag except zero, the autocovarince function of white noise is zero by independence • At lag zero, the autocovariance of white noise is just its variance, equal to one cov [wn(t)*wn(t)] = E[wn(t)*wn(t)] =1

22. Independence: Conceptual • the autocovariance function can be standardized, i,e, made free of units or scale, by dividing by the variance to obtain the autocorrelation function, symbolized for wn(t) by rwn, wn (u) = cov [wn(t)*wn(t-u)/Var wn(t) • In general, the autocorrelation function for a time series depends both on time, t, and lag, u. However, for stationary time series it depends only on lag.

23. Theoretical Autocorrelation Function: White Noise

24. What use is the autocorrelation function in practice? • Estimated Autocorrelations in EViews

25. 1000 observations of White Noise

26. Analysis • Breaking down the structure of an observed time series, i.e. modeling it • Example: weekly closing price of gold, Handy & Harmon, \$ per ounce

27. PRICE OF GOLD

28. Price of gold does Not look like white noise

29. What now? • How about week to week changes in the price of gold? • In EViews: Gen dgold = gold –gold(-1)

30. Changes in the price of gold • If changes in the price of gold are not significantly different from white noise, then we have a use for our white noise model: dgold(t) = c + wn(t) • Ignore the constant for the moment • What sort of time series is the price of gold?

31. The price of gold • dgold(t) = gold(t) – gold(t-1) = wn(t) • i.e. gold(t) = gold(t-1) + wn(t) • Lag by one: dgold(t-1) = gold(t-1) – gold(t-2) =wn(t-1) • i.e., gold (t-1) = gold(t-2) + wn (t-1), so gold(t) = wn(t) + wn(t-1)+ gold(t-2)

32. The price of gold • Keep lagging and substituting, and • gold(t) = wn(t) + wn(t-1) + wn(t-2) + …. • i.e. the price of gold is the current shock, wn(t), plus last week’s shock, wn(t-1), plus the shock from the week before that, wn(t-2) etc. • These shocks are also called innovations

33. The price of gold • This time series for gold, i.e. the sum of current and previous shocks is called a random walk, rw(t) • So rw(t) = wn(t) + wn(t-1) + wn(t-2) + … • Lagging by one: • rw(t-1) = wn(t-1) + wn(t-2) + wn(t-3) + … • So drw(t) = rw(t) –rw(t-1) = wn(t)

34. The first difference of a random walk • The first difference of a random walk is white noise

35. Random walk plus trend • If the price of gold is trend plus a random walk: gold(t) = a + b*t + rw(t), it is said to be a random walk with drift • Lagging by one, gold(t-1) = a + b*(t-1) + rw(t-1) • And subtracting, dgold(t) = b + drw(t), i.e. • dgold(t) = constant + white noise

36. The time series is too short for the constant To be significant

37. Simulated Random walk • EViews, sample 1 1, gen rw = wn • Sample 2 1000, gen rw = rw(-1) + wn

38. Simulated random walk