1 Power 2 Econ 240C

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

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6 Data in Excel YearGDP_CANGDP_CAN(-1C_CAN

7Stacking So for stacking, the data start with 1951

8 Data in Excel yearGDP_CANGDP_CAN(-1)C_CAN

9Stacking 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

10Stacking 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

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13Outline 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)

14 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

15 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 + c 1 *Q 1 (t) + c 2 *Q 2 (t) + c 3 *Q 3 (t) + e(t)

16 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?

17 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

18 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

19 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

20 The first ten values of simulated white noise, wn(t) Valuedraw = time index

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

22 Histogram and Statistics, 1000 Obs.

23Independence 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

24 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

25 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

26 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

27 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  wn, 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.

28 Theoretical Autocorrelation Function: White Noise

29 What use is the autocorrelation function in practice? Estimated Autocorrelations in EViews

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observations of White Noise

32Analysis Breaking down the structure of an observed time series, i.e. modeling it Example: weekly closing price of gold, Handy & Harmon, $ per ounce

33 PRICE OF GOLD DateWeekPrice 4/16/040$ /23/041$ /30/042$ /07/043$ /13/044$ /20/045$385.30

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36 Price of gold does Not look like white noise

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

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42 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?

43 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)

44 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

45 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)

46 The first difference of a random walk The first difference of a random walk is white noise

47 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

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

49 Simulated Random walk EViews, sample 1 1, gen rw = wn Sample , gen rw = rw(-1) + wn

50 Simulated random walk timeWhite noiseRandom walk

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54 Random walk Is a random walk evolutionary or stationary?

55 Random walk Mean function for a random walk, m(t) m(t) = E[rw(t)] = E[ wn(t) +wn(t-1) + …] m(t) = ….= 0

56 Variance of an infinite rw(t) Var[rw(t)] = E[rw(t)*rw(t)] Var[rw(t)] =E{[wn(t) + wn(t-1) + wn(t-2) …]*[wn(t) + wn(t-1) + wn(t-2) ….] Var rw(t) =        ∞ So the variance of an infinitely long random walk is not bounded, but infinite, and a random walk can go wandering off.

57 Random walk model The price of gold is bounded below by zero and is not likely to go wandering off to infinity either, so the random walk model is an approximation for the price of gold.

58Question What does the autocovariance function of an infinite random walk look like plotted against lag? lag 0  rw, rw

59 Recall the autocorrelation function For a finite sample of a simulated Random walk decays slowly

60Summary We are now familiar with two time series, white noise and random walks We have looked at the theoretical autocorrelation functions, or are in the process of doing so. We have simulated sample of both and looked at their empirically estimated autocorrelation functions, benchmarks for identification