动态经济计量模型
与时间序列模型
罗 凯
2005.11.22
Notice
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为防止丢失，上机作业统一交至助教 黄
国华师兄 信箱
huangguohua@gsm.pku.edu.cn
动态经济计量模型
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时间
静态模型－－同期影响
yt  0  1 xt  ut , t  1, 2,..., T
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分布滞后模型－－持续影响
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分布滞后模型
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无约束有限分布滞后模型
q
yt  c0   i xt i  ut ,
t  1, 2,..., T
i 0
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困难：
时间序列期数有限，q会过多占用自由度；
et自相关会很严重；
多重共线性
分布滞后模型
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滞后期长度
Adjusted R－square
AIC准则：
e ' e 2q
ln
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T
T
分布滞后模型
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例如：两期滞后模型
yt  0  1 xt  2 xt 1  3 xt 2  ut ,
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Reg yt xt xtlag1 xtlag2
看Adjusted R－square
进行序贯F 检验
t  3,..., T
分布滞后模型－－方法（1）
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多项式分布滞后模型
q
yt  c0   i xt i  ut ,
t  1, 2,..., T
i   0  1i   2i 2   3i 3
i  0,..., q
i 0
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步骤：
1.先定义新的解释变量：
Zt0=xt+xtlag1+xtlag2+…+xtlagq
分布滞后模型
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Zt1= xtlag1+2xtlag2+…+qxtlagq
Zt2= xtlag1+4xtlag2+…+q^2xtlagq
Zt3= xtlag1+8xtlag2+…+q^3xtlagq
2.run OLS
reg yt zt0-zt3
3. 根据下式，回求  0 ,...,  q
i   0  1i   2i 2   3i 3
i  0,..., q
分布滞后模型－－方法（2）
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cnsreg -- Constrained linear regression
cnsreg depvar indepvars [if] [in] [weight] , constraints(constraints)
[options]
options
description
-------------------------------------------------------------------------Model
* constraints(constraints) apply specified linear constraints
noconstant
suppress constant term
SE/Robust
vce(vcetype)
vcetype may be bootstrap or jackknife
分布滞后模型－－方法（2）
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Reporting
level(#)
set confidence level; default is level(95)
-------------------------------------------------------------------------* constraints(constraints) is required.
depvar and indepvars may contain time-series operators; see tsvarlist.
bootstrap, by, jackknife, rolling, statsby, and xi are allowed; see
prefix.
aweights and fweights are allowed; see weight.
See cnsreg postestimation for features available after estimation.
Description
cnsreg fits constrained linear regression models. cnsreg typed without
arguments redisplays the previous cnsreg results.
分布滞后模型－－方法（2）
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Options
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+-------+
----+ Model +-------------------------------------------------------------
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constraints(constraints), noconstant; see estimation options.
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+-----------+
----+ SE/Robust +---------------------------------------------------------
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vce(vcetype); see vce_option.
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+-----------+
----+ Reporting +--------------------------------------------------------level(#); see estimation options.
分布滞后模型－－方法（2）
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Examples
. constraint define 1 price = weight
. cnsreg mpg price weight, constraints(1)
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constraint def 1 price = weight
constraint def 2 displ = weight
constraint def 3 gear_ratio = -foreign
cnsreg mpg price weight displ gear_ratio foreign length, c(1-3)
predict mpghat if e(sample)
. constraint define 99 _cons = 0
. cnsreg mpg price weight displ gear_ratio foreign length, c(1-3,99)
分布滞后模型－－方法（2）
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如果原模型有q个滞后，则约束的个数为
q－p个。接上例，进一步设滞后5期，因
p＝3，因此，有2个约束。
依次为：
0  41  62  43  4  0
1  42  63  4 4  5  0
几何滞后模型 －自回归形式
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方法一：
var -- Vector autoregression models
var depvarlist [if] [in] [, options]
Var yt xt ylag
方法二：
prais depvar [indepvars] [if] [in] [, options]
此外，还是可以试着run一下：FGLS，
xtivreg
几何滞后模型 －移动平均形式
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主要方法：
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非线性最小二乘法
Stata 语句示例
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NLS语句
nl fcn depvar [varlist] [weight][if
exp][in range] [,level(#)] init(…)
lnlsq(#) leave eps(#) nolog trace
iterate(#) delta(#) fcn_options]
nlinit # parameter_list.(给参数赋初值).
predictnl yhat =将参数最终估计值带入
的回归方程式;
Stata 常用的函数(系统已内设)
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Exponential regression with one asymptote:
nl exp3
Y = b0 + b1*b2^X
nl exp2
Y=
b1*b2^X
nl exp2a Y =
b1*(1-b2^X)
Logistic function (symmetric sigmoid shape)(*):
nl log4
Y = b0 + b1/(1 + exp(-b2*(X-b3)))
nl log3
Y=
b1/(1 + exp(-b2*(X-b3)))
Gompertz function (asymmetric sigmoid shape):
nl gom4
Y = b0 + b1*exp(-exp(-b2*(X-b3)))
nl gom3
Y=
b1*exp(-exp(-b2*(X-b3)))
函数编程示例
program nlfcn
version 8.0 if "`1'"=="?"
{ global S_1 "parameter names" (initialize
parameters)
exit }
replace `1'= ... end
注意:具体函数名称前面的nl与函数名称中
间无空格，且不可去掉.以后可以直接调
用，注意语句格式:nl fcn depvar
indepvars
动态回归模型
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ARMAX方法
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或：(差分法)
动态回归模型
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arima -- ARIMA, ARMAX, and other dynamic regression models
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Basic syntax for a regression model with ARMA disturbances
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arima depvar [indepvars], ar(numlist) ma(numlist)
Basic syntax for an ARIMA(p,d,q) model
arima depvar, arima(#p,#d,#q)
Basic syntax for a multiplicative seasonal ARIMA(p,d,q)*(P,D,Q)s model
arima depvar, arima(#p,#d,#q) sarima(#P,#D,#Q,#s)
Full syntax
arima depvar [indepvars] [if] [in] [weight] [, options]
动态回归模型
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options
description
-------------------------------------------------------------------------Model
noconstant
suppress constant term
arima(#p,#d,#q)
specify ARIMA(p,d,q) model for dependent
variable
ar(numlist)
autoregressive terms of the structural model
disturbance
ma(numlist)
moving-average terms of the structural model
disturbance
Model 2
constraints(constraints) apply specified linear constraints
sarima(#P,#D,#Q,#s)
specify period-#s multiplicative seasonal
ARIMA term
mar(numlist, #s)
multiplicative seasonal autoregressive
terms; may be repeated
mma(numlist, #s)
multiplicative seasonal moving-average
terms; may be repeated
动态回归模型
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Model 3
condition
savespace
diffuse
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state(#|matname)
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p0(#|matname)
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SE/Robust
vce(vcetype)
robust
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Reporting
level(#)
detail
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Max options
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use conditional MLE instead of full MLE
conserve memory during estimation
use diffuse prior for starting Kalman filter
recursions
use alternate state vector for starting
Kalman filter recursions
use alternate prior for starting Kalman
recursions; seldom used
vcetype may be opg, robust, or oim
synonym for vce(robust)
set confidence level; default is level(95)
report list of gaps in time series
动态回归模型
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maximize_options
control the maximization process; seldom
used
-------------------------------------------------------------------------You must tsset your data before using arima; see tsset.
depvar and indepvars may contain time-series operators; see tsvarlist.
by, rolling, statsby, and xi may be used with arima; see prefix.
iweights are allowed; see weights.
See arima postestimation for features available after estimation.
Description
arima fits univariate models with time-dependent disturbances. arima fits
a model of depvar on indepvars where the disturbances are allowed to
follow a linear autoregressive moving-average (ARMA) specification. The
dependent and independent variables may be differenced or seasonally
differenced to any degree. When independent variables are included in the
specification, such models are frequently called ARMAX models; and when
independent variables are not specified, they reduce to Box-Jenkins
autoregressive integrated moving-average (ARIMA) models in the dependent
动态回归模型
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variable. Multiplicative seasonal ARIMA and ARMAX models can also be
fitted. Missing data are allowed and are handled using the Kalman filter
and methods outlined in [TS] arima.
In the full syntax, depvar is the variable being modeled, and the
structural or regression part of the model is specified in indepvars.
ar() and ma() specify the lags of autoregressive and moving-average terms,
respectively; and mar() and mma() specify the multiplicative seasonal
autoregressive and moving-average terms, respectively.
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arima allows time-series operators in the dependent variable and
independent variable lists, and it is often convenient to make extensive
use of these operators; see dates for an extended discussion of
time-series operators.
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arima typed without arguments redisplays the previous estimates.
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Options
动态回归模型
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+-------+
----+ Model +-------------------------------------------------------------
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noconstant; see estimation options.
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arima(#p,#d,#q) is an alternative, shorthand notation for specifying
models with ARMA disturbances. The dependent variable and any
independent variables are differenced #d times, 1 through #p lags of
autocorrelations and 1 through #q lags of moving averages are included
in the model. For example, the specification
. arima D.y, ar(1/2) ma(1/3)
is equivalent to
. arima y, arima(2,1,3)
The latter is easier to write for simple ARMAX and ARIMA models, but
if gaps in the AR or MA lags are to be modeled, of if different
operators are to be applied to independent variables, the first syntax
动态回归模型
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is required.
ar(numlist) specifies the autoregressive terms of the structural model
disturbance to be included in the model. For example, ar(1/3)
specifies that lags of 1, 2, and 3 of the structural disturbance be
included in the model; and ar(1 4) specifies that lags 1 and 4 be
included, perhaps to account for additive quarterly effects.
If the model does not contain regressors, these terms can also be
considered autoregressive terms for the dependent variable.
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ma(numlist) specifies the moving-average terms to be included in the
model. These are the terms for the lagged innovations (white-noise
disturbances).
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constraints(constraints); see estimation options for details.
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If constraints are placed between structural model parameters and ARMA
terms, the first few iterations may attempt steps into nonstationary
areas. This can be ignored if the final solution is well within the
动态回归模型
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bounds of stationary solutions.
+---------+
----+ Model 2 +----------------------------------------------------------sarima(#P,#D,#Q,#s) is an alternative, shorthand notation for specifying
the multiplicative seasonal components of models with ARMA
disturbances. The dependent variable and any independent variables
are lag-#s seasonally differenced #D times, and 1 through #P seasonal
lags of autoregressive terms and 1 through #Q seasonal lags of
moving-average terms are included in the model. For example, the
specification
. arima DS12.y, ar(1/2) mar(1/2,12) mma(1/2,12)
is equivalent to
. arima y, arima(2,1,3) sarima(2,1,2,12)
mar(numlist,#s) specifies the lag-#s multiplicative seasonal
autoregressive terms. For example, mar(1/2,12) requests that the
动态回归模型
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first two lag-12 multiplicative seasonal autoregressive terms be
included in the model.
mma(numlist,#s) specifies the lag-#s multiplicative seasonal
moving-average terms. For example, mma(1 3,12) requests that the
first and third (but not the second) lag-12 multiplicative seasonal
moving-average terms be included in the model.
+---------+
----+ Model 3 +----------------------------------------------------------condition specifies that conditional, rather than full, maximum likelihood
estimates be produced. This estimation method is not appropriate for
nonstationary series but may be preferable for long series or for
models that have one or more long AR or MA lags. diffuse, p0(), and
state0() may not be specified with condition. See [TS] arima for
details.
savespace specifies that memory use be conserved by retaining only those
variables required for estimation. The original dataset is restored
动态回归模型
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after estimation. This option is rarely used and should be used only
if there is insufficient space to fit a model without the option.
Note, however, that arima requires considerably more temporary storage
diffuse specifies that a diffuse prior be used as a starting point for the
during estimation than most estimation commands in Stata.
Kalman filter recursions. Using diffuse, nonstationary models may be
fitted with arima (see option p0() below; diffuse is equivalent to
specifying p0(1e9)). See [TS] arima for details.
state0(#|matname) is a rarely used option that specifies an alternate
initial state vector for starting the Kalman filter recursions. If #
is specified, all elements of the vector are taken to be #. The
default initial state vector is state0(0).
p0(#|matname) is a rarely specified option that can be used for
nonstationary series or when an alternate prior for starting the
Kalman recursions is desired; see [TS] arima for details.
+-----------+
动态回归模型
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----+ SE/Robust +---------------------------------------------------------
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vce(vcetype); see vce_option.
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robust; see estimation options.
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For state-space models in general and ARMAX and ARIMA models in
particular, the robust or quasi-maximum likelihood estimates (QMLE) of
variance are robust to symmetric non-normality in the disturbances,
including, as a special case, heteroskedasticity. The robust varianc
estimates are not generally robust to functional misspecification of
the structural or ARMA components of the model.
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+-----------+
----+ Reporting +---------------------------------------------------------
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level(#); see estimation options.
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detail specifies that a detailed list of any gaps in the series be
reported, including gaps due to missing observations or missing data
动态回归模型
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for the dependent variable or independent variables.
+-------------+
----+ Max options +------------------------------------------------------maximize_options: difficult, technique(algorithm_spec), iterate(#),
[no]log, trace, gradient, showstep, hessian, shownrtolerance,
tolerance(#), ltolerance(#), gtolerance(#), nrtolerance(#),
nonrtolerance(#), from(init_specs); see maximize.
These options are sometimes more important for ARIMA models than most
maximum likelihood models because of potential convergence problems
with ARIMA models, particularly if the specified model and the sample
data imply a nonstationary model.
Several alternate optimization methods, such as
Berndt-Hall-Hall-Hausman (BHHH) and Broyden-Fletcher-Goldfarb-Shanno
(BFGS), are provided for arima models. Although arima models are not
as difficult to optimize as ARCH models, their likelihoods are
nevertheless generally not quadratic and often pose optimization
动态回归模型
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difficulties; this is particularly true if a model is nonstationary or
nearly nonstationary. Since each method approaches optimization
differently, some problems can be successfully optimized by an
alternate method when one method fails.
The following options are all related to maximization and are
particularly important in fitting ARIMA models.
technique(algorithm_spec) specifies the optimization technique to use
to maximize the likelihood function.
technique(bhhh) specifies the Berndt-Hall-Hall-Hausman (BHHH)
algorithm.
technique(dfp) specifies the Davidon-Fletcher-Powell (DFP)
algorithm.
technique(bfgs) specifies the Broyden-Fletcher-Goldfarb-Shanno
(BFGS) algorithm.
动态回归模型
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technique(nr) specifies that Stata's modified Newton-Raphson (NR)
algorithm.
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You can specify multiple optimization methods. For example,
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technique(bhhh 10 nr 20)
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requests that the optimizer perform 10 BHHH iterations, switch to
Newton-Raphson for 20 iterations, switch back to BHHH for 10 more
iterations, and so on.
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The default for arima is technique(bhhh 5 bfgs).
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gtolerance(#) is a rarely used option that specifies a threshold for
the relative size of the gradient; see maximize. The default
gradient tolerance for arima is gtolerance(.05).
gtolerance(999) effectively disables the gradient criterion when
convergence is difficult to achieve. If the optimizer becomes
stuck with repeated "(backed up)" messages, the gradient probably
动态回归模型
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still contains substantial values, but an uphill direction cannot
be found for the likelihood. Using gtolerance(999) will often
obtain results but it may be unclear whether the global maximum
likelihood has been found. It is usually better to set the
maximum number of iterations (see maximize) to the point where the
optimizer appears to be stuck and then inspect the estimation
results.
from(init_specs) specifies the starting values of the model
coefficients; see maximize for a general discussion and syntax
options.
The standard syntax for from() accepts a matrix, a list of values,
or coefficient name value pairs; see maximize. In addition, arima
accepts from(armab0), which sets the starting value for all ARMA
paramters in the model to 0 prior to optimization.
ARIMA models may be sensitive to initial conditions and may have
coefficent values that correspond to local maxima. The default
starting values for arima are generally very good, particularly in
动态回归模型
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large samples for stationary series.
Examples
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. arima wpi, arima(1,1,1)
. arima D.wpi, ar(1) ma(1)
(same as above)
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. arima D.wpi, ar(1) ma(1 4)
(add quarterly MA effect)
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. arima wpi, arima(3,2,4)
. arima D2.wpi, ar(1/3) ma(1/4)
(ARIMA -- p=3, d=2, q=4)
(same as above)
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. arima lnair, arima(0,1,1) sarima(0,1,1,12)
ARIMA)
. arima DS12.lnair, ma(1) mma(1, 12)
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. arima consump m2 if tin( , 1981q4), ar(1) ma(1) robust
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(Multiplicative seasonal
(same as above)
时间序列模型
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单位根检验
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dfgls
DF-GLS unit-root test
dfuller
pperron
dwstat
durbina
bgodfrey
Augmented Dickey-Fuller unit-root test
Phillips-Perron unit-roots test
Durbin-Watson d statistic
Durbin's alternative test for serial correlation
Breusch-Godfrey test for higher-order serial
correlation
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时间序列模型
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archlm
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wntestb
wntestq
Engle's LM test for the presence of autoregressive
conditional heteroskedasticity
Bartlett's periodogram-based test for white noise
Portmanteau (Q) test for white noise
例：
Use c:\a.dta
tsset year
dfuller gdp (or: dfgls gdp 根据情况选)
时间序列模型
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Dickey-Fuller test for unit root
Number of obs =
20
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
6.995
-3.750
-3.000
-2.630
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 1.0000
发散。
不遵循平稳的AR(1)过程:[I(0)]，其差分也不是平稳的AR(1)过程:[I(1)]。
时间序列模型
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协整和误差修正模型
步骤：
1.run OLS reg y x (t)
2.predict abc, re
3.ge dabc=abc-abclag
4.dfuller dabc(or: dfgls dabc 根据情况选)
其他备用语句
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[TS] vecrank -- Estimate the cointegrating rank using Johansen's
framework
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Syntax
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vecrank depvar [if] [in] [, options]
options
description
-------------------------------------------------------------------------Model
lags(#)
use # for the maximum lag in underlying VAR
model
trend(constant)
include an unrestricted constant in model;
the
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trend(rconstant)
trend(trend)
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trend(rtrend)
trend(none)
default
include an restricted constant in model
include a linear trend in the cointegrating
equations and a quadratic trend in the
undifferenced data
include a restricted trend in model
do not include a trend or a constant model
Adv. model
sindicators(varlist_si) include normalized seasonal indicator
variables varlist_si
noreduce
do not perform checks and corrections for
collinearity among lags of dependent
variables
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Reporting
notrace
max
ic
level99
do not report of the trace statistic
report maximum-eigenvalue statistic
report information criteria
report 1% critical values instead of 5%
critical values
levela
report both 1% and 5% critical values
-------------------------------------------------------------------------You must tsset your data before using vecrank.
depvar may contain time-series operators; see tsvarlist.
by and rolling may be used with vecrank; see prefix.
vecrank does not allow gaps in the data.
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Description
vecrank produces statistics used to determine the number of cointegrating
equations in a vector error-correction model (VECM).
Options
+-------+
----+ Model +------------------------------------------------------------lags(#) specifies the number of lags in the VAR representation of the
model. The VECM will include one fewer lag of the first-differences.
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The number of lags must be greater than zero but small enough so that
the degrees of freedom used up by the model are less than the number
of observations.
trend(trend_spec) specifies one of five trend specifications to include in
the model. See vec intro and vec for descriptions. The default is
trend(constant).
+------------+
----+ Adv. model +-------------------------------------------------------sindicators(varlist_si) specifies normalized seasonal indicator variables
to be included in the model. The indicator variables specified in
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this option must be normalized. If the indicators are not properly
normalized, the likelihood-ratio-based tests for the number of
cointegrating equations do not converge to the asymptotic
distributions derived by Johansen. For details, see Methods and
Formulas of [TS] vec. sindicators() cannot be specified with
trend(none) or trend(rconstant).
noreduce causes vecrank to skip the checks and corrections for
collinearity among the lags of the dependent variables. By
default,
vecrank checks if the current lag specification causes some of the
regressions performed by vecrank to contain perfectly collinear
variables and reduces the maximum lag until the perfect
collinearity
is removed. See Collinearity in [TS] vec for more information.
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+-----------+
----+ Reporting +--------------------------------------------------------notrace requests that the output for the trace statistic not be displayed.
The default is to display the trace statistic.
max requests that the output for the maximum-eigenvalue statistic be
displayed. The default is to not display this output.
ic causes the output for the information criteria to be displayed. The
default is to not display this output.
level99 causes the 1% critical values to be displayed instead of the
default 5% critical values.
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levela causes both the 1% and the 5% critical values to be displayed.
Examples
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. vecrank y i c, lags(5)
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. vecrank y i c, lags(5) level99
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. vecrank y i c, lags(5) max levela notrace
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. vecrank y i c, lags(5) ic notrace