Stochastic Population Forecasting and ARIMA time series modelling Lectures QMSS Summer School, 2 July 2009

1. Stochastic Population ForecastingandARIMA time series modellingLectures QMSS Summer School, 2 July 2009 Nico Keilman Department of Economics, University of Oslo

2. Stochastic • Stochastic (from the Greek "Στόχος" for "aim" or "guess") means random. • A stochastic process is one whose behaviour is non-deterministic in that a system's subsequent state is determined both by the process's predictable actions and by a random element. • In a stochastic population forecast, uncertainty is made explicit: random variables are part of the forecast model.

3. Stochastic population forecast Future population / births / deaths /migrations as probability distributions, not one number (perhaps three)

4. Why Stochastic Population Forecasts (SPF)? Users should be informed about the expected accuracy of the forecast - probability of alternative future paths? - which forecast horizon is reasonable? Traditional deterministic forecast variants (e.g. High, Medium, Low) - do not quantify uncertainty  Prob(MediumPop) = 0 !! - give a misleading impression of uncertainty (example later) - leave room for politically motivated choices by the user

5. Outline • Uncertainty of population forecasts • Principles of SPF • Time series models (selected examples) • Alho’s scaled model for error • Examples from UPE • Using a SPF Focus on national forecasts

6. How uncertain are population forecasts? Empirical findings – historical forecasts evaluated against actual population numbers (ex post facto)

7. Main findings for official forecasts in Western countries • Uncertainty in forecasts of certain population variables surprisingly large • Forecasts for the young and the old age groups are the least reliable • Forecast errors increase as forecast interval lengthens • Large uncertainty for small countries • Large uncertainty for countries that are strongly affected by migration • European forecasts have not become more accurate since WW2

8. Errors in age structure forecasts Europe

9. United Kingdom - men

10. United Kingdom - women

11. Why uncertain? • Data quality (LDC’s) • Social science predictions, no accurate behavioural theory • Rely on observed regularities instead  Problems when sudden trend shifts occur - stagnation life expectancy men 1950s - baby boom/baby bust

12. Traditional population forecasts do not give a correct impression of uncertainty

13. Example: Old Age Dependency Ratio (OADR) for Norway in 2060Source: Statistics Norway population forecast of 2005 High Middle Low |H-L|/M millions (%) POP67+ 1.55 1.33 1.13 31 POP20-66 4.03 3.39 2.83 36 OADR 0.38 0.39 0.40 4

14. Two major problems • Wide margins for some variables, narrow margins for others • Narrow margins in the short run,wide margins in the long run - implicitly assumed perfect autocorrelation (and sometimes perfect correlation across components)

15. Coverage probabilities for H-L margin of total population in official forecasts 2010 2050 Statistics Norway 47% 78% Statistics Sweden • Fertility 19% 32% • Mortality 4% 20% • Migration 1% 34% Sources: Stochastic population forecasts from UPE Traditional forecasts from Statistics Norway and Statistics Sweden

17. Cohort-component method Deterministic population forecast Needed for the country in question: annual assumptions on future • Fertility  Total Fertility Rate • Mortality  Life expectancy at birth M/F • Migration  Net immigration • as well as rates (fertility, mortality) & numbers (migration) by age & sex

18. Stochastic Population Forecast: How? • Cohort-component method • Random rates for fertility and mortality, random numbers for net-migration • Normal distributions in the log scale (rates) or in the original scale (migration numbers) - expected values (“point predictions”) – cf. Medium variant in traditional deterministic forecast - standard deviations - correlations (age, time, sex, components, countries)

19. SPF: How? (cntnd) • Joint distribution of all random input variables (rates, migration numbers) • In practice: simplifications, e.g. - independence of components (fertility, mortality, migration) - correlation between male and female mortality (constant across ages, time) • One random draw from all prob. distributions  one sample path • Repeated draws  thousands of sample paths

20. SPF: How? (cntnd) Three main approaches: uncertainty parameters based on historical errors expert knowledge statistical model

21. SPF: Examples Multivariate time series models for all parameters of interest Examples for Norway 1995-2050, see http://folk.uio.no/keilman/6-15.pdf and European countries 2003-2050, see http://www.stat.fi/tup/euupe/index_en.html Alho’s scaled model for error, implemented in PEP (Program for Error Propagation) Example for aggregate of 18 European countries 2003-2050, see http://www.stat.fi/tup/euupe/index_en.html

22. Time series example, Norway:log(TFR) = ARIMA(1,1,0) Zt = 0.67Zt-1 + εt-1 , Zt = log(TFRt) - log(TFRt-1) (0.10)

23. Prediction intervals, age-specific fertility rates, Norway 2050

24. Time series models for • parameters of Gamma model for age-specific fertility (TFR, MAC, variance in age at childbearing) • e0 • parameters of Heligman-Pollard model for age-specific mortality • immigration numbers • emigration numbers (deterministic age patterns for both migration flows)  5000 simulations

25. Population size, Norway

26. Population size, Norway

27. Population size, Norway

28. Population size, Norway

29. Population size, Norway

30. Population size, Norway

31. Population size, Norway

32. Population size, Norway

33. Population size, Norway

35. Time series models, two examples 1. Autoregressive model of order 1 - AR(1) Zt = φZt-1 + εt |φ| < 1, εt i.i.d random variables, zero expectation, constant variance – ”white noise” Var(Zt) = Var(εt)(1- φ2t)/(1- φ2) constant (in the long run – large t) For large t: k-step ahead autocorrelation Corr(Zt, Zt+k) equals φk , independent of time

36. 2. Random Walk - RW Zt = Zt-1 + εt Var(Zt) = t*Var(εt) unbounded for large t Independent increments (zero autocorrelation)

37. Forecasts and 95% prediction intervals for net migration. Data 1960-2000 Outliers: 1989 AR(1) & const:Zt=5688+0.76Zt-1+εtOutliers: 1962, 1988AR(1) & const:Zt=7819+0.39Zt-1+εt

38. Forecasts and 67%, 80%, and 95% prediction intervals for the TFR. Data 1950-2000. Observed TFR-value for the year 2000 is given as “y2000” Model: AR(1) & constant Zt (=logTFRt) = 0.001 + 0.988Zt-1 + εt

39. Forecasts and 67%, 80%, and 95% prediction intervals for the TFR. Data 1900-2000. Observed TFR-value for the year 2000 is given as “y2000” Model: AR(1) & constant Outliers 1920, 1942Zt (=logTFRt) = -0.003 + 0.995Zt-1 + εt

40. Forecasts and 67%, 80%, and 95% prediction intervals for the TFR. Data 1950-2000. Observed TFR-value for the year 2000 is given as “y2000” Model: AR(2) & constant Zt (=logTFRt) = 0.002 + 0.941Zt-1 - 0.408Zt-2 + εt

41. Forecasts and 67%, 80%, and 95% prediction intervals for the TFR. Data 1900-2000. Observed TFR-value for the year 2000 is given as “y2000” Model: AR(2)-ARCH(1) Outliers 1919, 1920, 1940, 1941Zt (=logTFRt) = 0.005 + 0.981Zt-1 + vt + dummiesvt = 0.214 vt-2 + εt,εt = (√ht)et, ht = 7E-4+0.708(εt2)

42. Time series approach to SPF + conceptually simple - inflexible Alternative: Alho’s scaled model for error Implemented in Program for Error Propagation (PEP) http://www.joensuu.fi/statistics/software/pep/pepstart.htm .

43. Scaled model for error Suppose the true age-specific rate in age j during forecast year t > 0 is of the form R(j,t) = F(j,t)exp(X(j,t)), where F(j,t) is the point forecast, and X(j,t) is the relative error

44. Suppose that the error processes are of the form X(j,t) = ε(j,1) + ... + ε(j,t) with error increments of the form ε(j,t) = S(j,t)(ηj + δ(j,t)) S(j,t) deterministic scales. δ(j,t) are independent over time t. δ(j,t) are independent of ηj for all t and j ηj ~ N(0, κ), δ(j,t) ~ N(0, 1 - κ) , 0 ≤ κ ≤ 1 Note that Var(ε(j,t)) = S(j,t)2 A positive kappa means that there is systematic error in the time trend of the rate.

45. κ = Corr[ε(j,t), ε(j,t+h)] for all h > 0, thus κ is the (constant) autocorrelation between the error increments. Together, the autocorrelation κ and the scale S(j,t) determine the variance of the relative error X(j,t). Ex. 1. Under a random walk model the error increments are uncorrelated with κ = 0. Ex. 2. The model with constant scales (S(j,t)=S(j)) can be interpreted as a random walk with a random drift. The relative importance of the two components is determined by κ.

46. Migration Migration (net) is represented in absolute terms Dependence on age is deterministic, given by a fixed distribution g(j,x) over age x The error of net migration in age x, for sex j, during year t > 0, is additive and of the form Y(j,x,t) = S(j,t)g(j,x)(ηj + δ(j,t))

47. Key properties of the scaled model • The choice of the scales S(j,t) is unrestricted. Hence any sequence of non-decreasing error variances can be matched (e.g. heteroscedasticity) • Any sequence of cross-correlations over ages can be majorized using the AR(1) models of correlation • Any sequence of autocorrelations for the error increments can be majorized.

48. Scaled model for error Used for UPE project: Uncertain Population of Europe • 18 countries: EU15 + Iceland, Norway, Switzerland (EEA+) • 2003 – 2050 • Probability distributions specified on the basis of - time series analysis (TFR, e0, net-migr.) - empirical forecast errors - expert judgement • 3000 simulations for each country, PEP • http://www.stat.fi/tup/euupe/index_en.html

49. Population size EEA+median (black), 80% prediction intervals (red) 77% chance > 400 million in 2050 (UN)83% chance > 392 million in 2050 (2003)

50. median (black), 80% prediction intervals (red)