Kinh tế lượng ứng dụng

đổi tuyệt đối của Y là 0,012. iii uXY  ln21  X dX dY 2 4.4.2. Mô hình lin-log        XdX dY 1 2 hay 16 Ví dụ Y: GNP (tỷ USD) X: lượng cung tiền (tỷ USD) Với số liệu trong khoảng thời gian 1970-83 Ý nghĩa 2=2584,785: trong khoảng thời gian 1970-83, lượng cung tiền tăng lên 1%, kéo theo sự gia tăng bình quân của GNP 25,84 tỷ USD. 4.4.2. Mô hình lin-log ii XY ln*785,258421,16329 ˆ  17 Đặc điểm: Khi X tiến tới ∞, số hạn β2(1/X) tiến dần tới 0 và Y tiến tới giá trị tới hạn β1. Ứng dụng: đường chi phí đơn vị, đường tiêu dùng theo thu nhập Engel hoặc đường cong Philip. ii u X Y  1 21  4.5 Mô hình nghịch đảo 18 Chi phí sản xuất cố định trung bình (AFC) giảm liên tục khi sản lượng tăng và cuối cùng tiệm cận với trục sản lượng ở β1 1 >0 2 >0 1 X (sản lượng) Y (AFC) 0 Đường chi phí đơn vị 19 Khi tỷ lệ thất nghiệp tăng vô hạn, tỷ lệ giảm sút của tiền lương sẽ không vượt quá β1 1 <0 2 >0 1 X (Tỷ lệ thất nghiệp) Y (Tỷ lệ thay đổi tiền lương) 0 Đường cong Phillips 20 1 > 0 2 < 0 1 X (Tổng thu nhập/ Tổng chi tiêu) Y (Chi tiêu của một loại hàng) 0 -2 / 1 Đường cong Engel 21 Chi tiêu hàng hóa tăng khi tổng thu nhập (hoặc tổng chi tiêu) tăng nhưng đối với một số loại hàng hóa thì thu nhập của người tiêu dùng phải đạt ở mức tối thiểu -2 / 1 (hay còn gọi là ngưỡng thu nhập) thì người tiêu dùng mới sử dụng loại hàng này. Mặt khác, nhu cầu của loại hàng này là hữu hạn, nghĩa là dù thu nhập có tăng vô hạn thì người tiêu dùng cũng không tiêu thụ thêm mặt hàng này nữa. Mức tiêu dùng bão hòa của loại hàng này là β1 Đường cong Engel 22 Với: Y Tổng chi phí X Số lượng sản phẩm Ứng dụng: từ hàm này, suy ra được chi phí trung bình (AC) và chi phí biên (MC) ii uXXXY  3 4 2 321  4.6 Mô hình đa thức 23 Với: Yt Tiêu dùng năm t Xt Thu nhập năm t Xt-1 Thu nhập năm t-1 Xt-k Thu nhập năm t-k k Chiều dài độ trễ tktttt uXXXY   41321 ...  4.7 Mô hình có độ trễ phân phối 24 So sánh R2 giữa các mô hình Cùng cỡ mẫu n Cùng số biến độc lập. Nếu các hàm hồi quy không cùng số biến độc lập thì dùng hệ số xác định hiệu chỉnh Biến phụ thuộc xuất hiện trong hàm hồi quy có cùng dạng. Biến độc lập có thể ở các dạng khác nhau. VD: Các hàm hồi quy có thể so sánh R2 với nhau Y=β1 + β.X +U Y= β1 + β.lnX +U Các hàm hồi quy không thể so sánh R2 với nhau Y=β1 + β.X +U lnY= β1 + β.X +U 2 R Copyright 1996 Lawrence C. Marsh Time Series Analysis Copyright 1996 Lawrence C. Marsh Previous Chapters used Economic Models 1. economic model for dependent variable of interest. 2. statistical model consistent with the data. 3. estimation procedure for parameters using the data. 4. forecast variable of interest using estimated model. Times Series Analysis does not use this approach. Copyright 1996 Lawrence C. Marsh Time Series Analysis is useful for short term forecasting only. Time Series Analysis does not generally incorporate all of the economic relationships found in economic models. Times Series Analysis uses more statistics and less economics. Long term forecasting requires incorporating more involved behavioral economic relationships into the analysis. Copyright 1996 Lawrence C. Marsh Univariate Time Series Analysis can be used to relate the current values of a single economic variable to: 1. its past values 2. the values of current and past random errors Other variables are not used in univariate time series analysis. Copyright 1996 Lawrence C. Marsh 1. autoregressive (AR) 2. moving average (MA) 3. autoregressive moving average (ARMA) Three types of Univariate Time Series Analysis processes will be discussed in this chapter: Copyright 1996 Lawrence C. Marsh 1. its past values. 2. the past values of the other forecasted variables. 3. the values of current and past random errors. Multivariate Time Series Analysis can be used to relate the current value of each of several economic variables to: Vector autoregressive models discussed later in this chapter are multivariate time series models. Copyright 1996 Lawrence C. Marsh First-Order Autoregressive Processes, AR(1): yt = d + q1yt-1+ et, t = 1, 2,...,T. (16.1.1) d is the intercept. q1 is parameter generally between -1 and +1. et is an uncorrelated random error with mean zero and variance se 2 . Copyright 1996 Lawrence C. Marsh Autoregressive Process of order p, AR(p) : yt = d + q1yt-1 + q2yt-2 +...+ qpyt-p + et (16.1.2) d is the intercept. qi’s are parameters generally between -1 and +1. et is an uncorrelated random error with mean zero and variance se 2 . Copyright 1996 Lawrence C. Marsh AR models always have one or more lagged dependent variables on the right hand side. Consequently, least squares is no longer a best linear unbiased estimator (BLUE), but it does have some good asymptotic properties including consistency. Properties of least squares estimator: Copyright 1996 Lawrence C. Marsh AR(2) model of U.S. unemployment rates yt = 0.5051 + 1.5537 yt-1 - 0.6515 yt-2 (0.1267) (0.0707) (0.0708) Note: Q1-1948 through Q1-1978 from J.D.Cryer (1986) see unempl.dat positive negative Copyright 1996 Lawrence C. Marsh Choosing the lag length, p, for AR(p): The Partial Autocorrelation Function (PAF) The PAF is the sequence of correlations between (yt and yt-1), (yt and yt-2), (yt and yt-3), and so on, given that the effects of earlier lags on yt are held constant. Copyright 1996 Lawrence C. Marsh Partial Autocorrelation Function yt = 0.5 yt-1 + 0.3 yt-2 + et 0 2 / T - 2 / T 1 -1 k qkk is the last (k th) coefficient in a kth order AR process. This sample PAF suggests a second order process AR(2) which is correct. Data simulated from this model: qkk ^ Copyright 1996 Lawrence C. Marsh Using AR Model for Forecasting: unemployment rate: yT-1 = 6.63 and yT = 6.20 yT+1 = d + q1 yT + q2 yT-1 = 0.5051 + (1.5537)(6.2) - (0.6515)(6.63) = 5.8186 ^ ^ ^ ^ yT+2 = d + q1 yT+1 + q2 yT = 0.5051 + (1.5537)(5.8186) - (0.6515)(6.2) = 5.5062 ^ ^ ^ ^ yT+1 = d + q1 yT + q2 yT-1 = 0.5051 + (1.5537)(5.5062) - (0.6515)(5.8186) = 5.2693 ^ ^ ^ ^ Copyright 1996 Lawrence C. Marsh Moving Average Process of order q, MA(q): yt = m + et + a1et-1 + a2et-2 +...+ aqet-q + et (16.2.1) m is the intercept. ai‘s are unknown parameters. et is an uncorrelated random error with mean zero and variance se 2 . Copyright 1996 Lawrence C. Marsh An MA(1) process: yt = m + et + a1et-1 (16.2.2) Minimize sum of least squares deviations: S(m,a1) = S et = S(yt - m - a1et-1) (16.2.3) 2 t=1 T t=1 T 2 Copyright 1996 Lawrence C. Marsh stationary: A stationary time series is one whose mean, variance, and autocorrelation function do not change over time. nonstationary: A nonstationary time series is one whose mean, variance or autocorrelation function change over time. Stationary vs. Nonstationary Copyright 1996 Lawrence C. Marsh Copyright 1996 Lawrence C. Marsh Copyright 1996 Lawrence C. Marsh Copyright 1996 Lawrence C. Marsh Copyright 1996 Lawrence C. Marsh NONSTATIONARY PROCESSES 10 -15 -10 -5 0 5 10 15 20 1 11 21 31 41 51 61 71 81 91 The chart shows a typical random walk. If it were a stationary process, there would be a tendency for the series to return to 0 periodically. Here there is no such tendency. Random walk Copyright 1996 Lawrence C. Marsh 16 STATIONARY PROCESSES Here is a series generated by this process with b2 = 0.7 and random numbers for the innovations. -15 -10 -5 0 5 10 15 20 1 11 21 31 41 51 61 71 81 91 Copyright 1996 Lawrence C. Marsh yt = z t - z t-1 First Differencing is often used to transform a nonstationary series into a stationary series: where z t is the original nonstationary series and yt is the new stationary series. Copyright 1996 Lawrence C. Marsh Choosing the lag length, q, for MA(q): The Autocorrelation Function (AF) The AF is the sequence of correlations between (yt and yt-1), (yt and yt-2), (yt and yt-3), and so on, without holding the effects of earlier lags on yt constant. The PAF controlled for the effects of previous lags but the AF does not control for such effects. Copyright 1996 Lawrence C. Marsh Autocorrelation Function yt = et - 0.9 et-1 0 2 / T - 2 / T 1 -1 k rkk rkk is the last (k th) coefficient in a kth order MA process. This sample AF suggests a first order process MA(1) which is correct. Data simulated from this model: Copyright 1996 Lawrence C. Marsh Autoregressive Moving Average ARMA(p,q) An ARMA(1,2) has one autoregressive lag and two moving average lags: yt = d + q1yt-1 + et + a1et-1 + a2 et-2 Copyright 1996 Lawrence C. Marsh Integrated Processes A time series with an upward or downward trend over time is nonstationary. Many nonstationary time series can be made stationary by differencing them one or more times. Such time series are called integrated processes. Copyright 1996 Lawrence C. Marsh The number of times a series must be differenced to make it stationary is the order of the integrated process, d. An autocorrelation function, AF, with large, significant autocorrelations for many lags may require more than one differencing to become stationary. Check the new AF after each differencing to determine if further differencing is needed. Copyright 1996 Lawrence C. Marsh Unit Root zt = q1zt -1 + m + et + a1et -1 (16.3.2) -1 < q1 < 1 stationary ARMA(1,1) q1 = 1 nonstationary process q1 = 1 is called a unit root Copyright 1996 Lawrence C. Marsh Unit Root Tests Dzt = q1zt -1 + m + et + a1et -1 (16.3.3) Testing q1 = 0 is equivalent to testing q1 = 1 zt - zt -1 = (q1- 1)zt -1 + m + et + a1et -1 * where Dzt = zt - zt -1 and q1 = q1- 1 * * Copyright 1996 Lawrence C. Marsh Unit Root Tests H0: q1 = 0 vs. H1: q1 < 0 (16.3.4) * * Computer programs typically use one of the following tests for unit roots: Dickey-Fuller Test Phillips-Perron Test Copyright 1996 Lawrence C. Marsh Autoregressive Integrated Moving Average ARIMA(p,d,q) An ARIMA(p,d,q) model represents an AR(p) - MA(q) process that has been differenced (integrated, I(d)) d times. yt = d + q1yt-1 +...+ qpyt-p + et + a1et-1 +... + aq et-q Copyright 1996 Lawrence C. Marsh The Box-Jenkins approach: 1. Identification determining the values of p, d, and q. 2. Estimation linear or nonlinear least squares. 3. Diagnostic Checking model fits well with no autocorrelation? 4. Forecasting short-term forecasts of future yt values. Copyright 1996 Lawrence C. Marsh Vector Autoregressive (VAR) Models yt = q0 + q1yt-1 +...+ qpyt-p + f1xt-1 +... + fp xt-p + et xt = d0 + d1yt-1 +...+ dpyt-p + a1xt-1 +... + ap xt-p + ut Use VAR for two or more interrelated time series: Copyright 1996 Lawrence C. Marsh 1. extension of AR model. 2. all variables endogenous. 3. no structural (behavioral) economic model. 4. all variables jointly determined (over time). 5. no simultaneous equations (same time). Vector Autoregressive (VAR) Models Copyright 1996 Lawrence C. Marsh The random error terms in a VAR model may be correlated if they are affected by relevant factors that are not in the model such as government actions or national/international events, etc. Since VAR equations all have exactly the same set of explanatory variables, the usual seemingly unrelation regression estimation produces exactly the same estimates as least squares on each equation separately. Copyright 1996 Lawrence C. Marsh Consequently, regardless of whether the VAR random error terms are correlated or not, least squares estimation of each equation separately will provide consistent regression coefficient estimates. Least Squares is Consistent Copyright 1996 Lawrence C. Marsh VAR Model Specification To determine length of the lag, p, use: 2. Schwarz’s SIC criterion 1. Akaike’s AIC criterion Copyright 1996 Lawrence C. Marsh Spurious Regressions yt = b1 + b2 xt + et where et = q1 et-1 + nt -1 < q1 < 1 I(0) (i.e. d=0) q1 = 1 I(1) (i.e. d=1) If q1 =1 least squares estimates of b2 may appear highly significant even when true b2 = 0 . Copyright 1996 Lawrence C. Marsh Cointegration yt = b1 + b2 xt + et If xt and yt are nonstationary I(1) we might expect that et is also I(1). However, if xt and yt are nonstationary I(1) but et is stationary I(0), then xt and yt are said to be cointegrated. Copyright 1996 Lawrence C. Marsh Cointegrated VAR(1) Model yt = q0 + q1yt-1 + f1xt-1 + et xt = d0 + d1yt-1 + a1xt-1 + ut VAR(1) model: If xt and yt are both I(1) and are cointegrated, use an Error Correction Model, instead of VAR(1). Copyright 1996 Lawrence C. Marsh Error Correction Model Dyt = q0 + (q1-1)yt-1 + f1xt-1 + et Dxt = d0 + d1yt-1 + (a1-1)xt-1 + ut Dyt = yt - yt-1 and Dxt = xt - xt-1 (continued) Copyright 1996 Lawrence C. Marsh Error Correction Model Dyt = q0 + g1(yt-1 - b1 - b2 xt-1) + et * Dxt = d0 + g2(yt-1 - b1 - b2 xt-1) + ut * q0 = q0 + g1b1 * d0 = d0 + g2b1 * g2 = d1 g1 = f1 d1 a1 - 1 b2 = d1 1 - a1 Copyright 1996 Lawrence C. Marsh yt-1 = b1 + b2 xt-1 + et-1 Estimate by least squares: to get the residuals: et-1 = yt-1 - b1 - b2 xt-1 ^ ^ ^ Estimating an Error Correction Model Step 1: Copyright 1996 Lawrence C. Marsh Estimate by least squares: Estimating an Error Correction Model Step 2: Dyt = q0 + g1 et-1 + et * Dxt = d0 + g2 et-1 + ut * ^ ^