R Hypothesis Test Notebook

R Hypothesis Test

Assume the predetermined significance level is 0.05. 假设预定的显着性水平是0.05。

1 Shapiro-Wilk Test W检验（Shapiro–Wilk (夏皮罗–威克尔 ) W统计量检验)

Null hypothesis: 零假设: The sample came from a normally distributed population. 样本来自正态分布总体

检验数据是否符合正态分布，R函数：shapiro.test().

结果含义：当p值小于某个显著性水平α(比如0.05)时，则认为 样本不是来自正态分布的总体，否则则承认样本来自正态分布的总体。

install.packages("stats")
library(stats)
# args() function displays the argument names and corresponding default values of a function or primitive.
# args()函数显示一个函数的参数名称和相应的默认值。
args(shapiro.test)
#function (x) 
#NULL

# Example 1:
# Test random sample from a normal distribution.
# 测试来自正态分布的随机抽样。
set.seed(1)
x <- rnorm(150)
res <- shapiro.test(x)

res$p.value # > 0.05
[1] 0.7885523
# Conclusion: We are unable to reject the null hypothesis.
# 结论：我们无法拒绝零假设。

# Example 2:
# Test daily observations of S&P 500 from 1981-01 to 1991-04.
# 测试S＆P500指数从1981-01到1991-04的日观察值。
install.packages("Ecdat")
library(Ecdat)
data(SP500)
class(SP500)
# [1] "data.frame"
SPreturn = SP500$r500 # use the $ to index a column of the data.frame
# 用$符号取出数据框中的一列
(res <- shapiro.test(SPreturn))
#         Shapiro-Wilk normality test
# data: SPreturn
# W = 0.8413, p-value < 2.2e-16

names(res)
# [1] "statistic" "p.value" "method" "data.name"

res$p.value # < 0.05
# [1] 2.156881e-46
# Conclusion: We should reject the null hypothesis.
# 结论：我们应该拒绝零假设。

2 Jarque-Bera Test

Null hypothesis: The skewness and the excess kurtosis of samples are zero. 样本的偏度和多余峰度均为零

install.packages("tseries")
library(tseries)
args(jarque.bera.test)
# function (x) 
# NULL

# Example 1: 
# Test random sample from a normal distribution
set.seed(1)
x <- rnorm(150)
res <- jarque.bera.test(x)

names(res)
# [1] "statistic" "parameter" "p.value" "method" "data.name"

res$p.value # > 0.05
# X-squared 
# 0.8601533 
# Conclusion: We should not reject the null hypothesis. 

# Example 2:
# Test daily observations of S&P 500 from 1981–01 to 1991–04
install.packages("Ecdat")
library(Ecdat)
data(SP500)
class(SP500)
# [1] "data.frame"
SPreturn = SP500$r500 # use the $ to index a column of the data.frame
(res <- jarque.bera.test(SPreturn))
#         Jarque Bera Test
# data: SPreturn
# X-squared = 648508.6, df = 2, p-value < 2.2e-16

names(res)
# [1] "statistic" "parameter" "p.value" "method" "data.name"

res$p.value # < 0.05
# X-squared 
#         0 
# Conclusion: We should reject the null hypothesis.

3 Correlation Test 相关性检验

R函数：cor.test()

Null hypothesis: The correlation is zero. 样本相关性为0

结果含义：如果p值很小，则拒绝原假设，认为x,y是相关的。否则认为是不相关的。

library(stats)
cor.test(x, y,
  alternative = c("two.sided", "less", "greater"),
  method = c("pearson", "kendall", "spearman"),
  exact = NULL, 
  conf.level = 0.95, ...)

另外一个例子：

install.packages("stats")
library(stats)
args(getS3method("cor.test","default"))
# function (x, y, alternative = c("two.sided", "less", "greater"), 
#     method = c("pearson", "kendall", "spearman"), exact = NULL, 
#     conf.level = 0.95, continuity = FALSE, ...) 
# NULL

# x, y: numeric vectors of the data to be tested
# x, y: 进行测试的数据的数值向量

# alternative: controls two-sided test or one-sided test
# alternative: 控制进行双侧检验或单侧检验

# method: "pearson", "kendall", or "spearman"

# conf.level: confidence level for confidence interval
# conf.level: 置信区间的置信水平


# Example:
# Test the correlation between the food industry and the market portfolio.
# 测试在食品行业的收益和市场投资组合之间的相关性。
data(Capm,package="Ecdat")
(res <- cor.test(Capm$rfood,Capm$rmrf))
#         Pearson's product-moment correlation
# 皮尔逊积矩相关
# data: Capm$rfood and Capm$rmrf
# t = 27.6313, df = 514, p-value < 2.2e-16
# alternative hypothesis: true correlation is not equal to 0
# 备择假设：真正的相关性不等于0
# 95 percent confidence interval:
# 95%置信区间
#  0.7358626 0.8056348
# sample estimates:
# 样本估计
#       cor 
# 0.7730767 

names(res)
# [1] "statistic" "parameter" "p.value" "estimate" 
# [5] "null.value" "alternative" "method" "data.name" 
# [9] "conf.int" 

res$p.value # < 0.05
# [1] 0
# Conclusion: We should reject the null hypothesis.

4 Durbin-Watson Test

Null hypothesis: The autocorrelation of the disturbance is 0. 干扰的自相关为0

install.packages("car")
library(car)
args(durbinWatsonTest) 
# dwt is an abbreviation for durbinWatsonTest
# dwt是durbinWatsonTest的简称。
# function (model, ...) 
# NULL
# model: a linear-model object, or a vector of residuals from a linear model
# model: 一个线性模型对象，或线性模式的残差向量

install.packages("lmtest")
library(lmtest)
args(dwtest)
# function (formula, order.by = NULL, alternative = c("greater", 
#     "two.sided", "less"), iterations = 15, exact = NULL, tol = 1e-10, 
#     data = list()) 
# NULL

# Example:
# Test autocorrelation in residuals of the CAPM model
# CAPM模型的残差自相关测试
fit.lm <- lm(Capm$rfood ~ Capm$rmrf)
(res <- dwt(fit.lm))
 # lag Autocorrelation D-W Statistic p-value
 #   1 0.1163382 1.765306 0.014
 # Alternative hypothesis: rho != 0

names(res)
# [1] "r" "dw" "p" "alternative"

res$p # < 0.05
# [1] 0.014
# Conclusion: We should reject the null hypothesis.

(res <- dwtest(fit.lm))
#         Durbin-Watson test
# data: fit.lm
# DW = 1.7653, p-value = 0.003757
# alternative hypothesis: true autocorrelation is greater than 0
# 备择假设：真正的自相关大于0

names(res)
# [1] "statistic" "method" "alternative" "p.value" 
# [5] "data.name" 

res$p.value # < 0.05
# [1] 0.003756644
# Conclusion: We should reject the null hypothesis.

5 Ljung-Box Test

Null hypothesis: The data are independently distributed, i.e. the autocorrelation coefficients are all zero.

数据是独立分布的，也就是说，它的自相关系数都是零

install.packages("stats")
library(stats)
args(Box.test)
# function (x, lag = 1, type = c("Box-Pierce", "Ljung-Box"), fitdf = 0) 
# NULL

install.packages("FinTS")
library(FinTS)
args(AutocorTest)
# function (x, lag = ceiling(log(length(x))), type = c("Ljung-Box", 
#     "Box-Pierce", "rank"), df = lag) 
# NULL

# Example:
# To test first n-lag autocorrelation coefficients are all zero.
# 测试前n次滞后自相关系数均为零。
(res <- Box.test(residuals(fit.lm),lag=5,type="Ljung-Box"))
        # Box-Ljung test
# data: residuals(fit.lm)
# X-squared = 13.7663, df = 5, p-value = 0.01716

names(res)
# [1] "statistic" "parameter" "p.value" "method" "data.name"

res$p.value # < 0.05
# [1] 0.01716429
# Conclusion: We should reject the null hypothesis.

(res <- AutocorTest(residuals(fit.lm),lag=5,type="Ljung-Box"))
        Box-Ljung test
# data: residuals(fit.lm)
# X-squared = 13.7663, df = 5, p-value = 0.01716

names(res)
# [1] "statistic" "parameter" "p.value" "method" 
# [5] "data.name" "Total.observ"

res$p.value # < 0.05
# [1] 0.01716429
# Conclusion: We should reject the null hypothesis.

6 Lagrange Multiplier Test

Null hypothesis: No ARCH effect. 无ARCH效应

install.packages("FinTS")
library(FinTS)
args(ArchTest)
# function (x, lags = 12, demean = FALSE) 
# NULL

# Example:
# Test the residuals for Autoregressive Conditional Heteroscedasticity (volatility clustering).
# 测试残差的自回归条件异方差（波动集聚性）。

(res <- ArchTest(residuals(fit.lm)))
        # ARCH LM-test; Null hypothesis: no ARCH effects
# data: residuals(fit.lm)
# Chi-squared = 182.0362, df = 12, p-value < 2.2e-16

names(res)
# [1] "statistic" "parameter" "p.value" "method" "data.name"

res$p.value # < 0.05
# Chi-squared 
          # 0 
# Conclusion: We should reject the null hypothesis.

7 Dickey-Fuller Test

Null hypothesis: There is a unit root. 单位根存在

install.packages("urca")
library(urca)
args(ur.df)

# function (y, type = c("none", "drift", "trend"), lags = 1, selectlags = c("Fixed", 
#     "AIC", "BIC")) 
# NULL
# y: time series
# y: 时间序列
# type: "none", "drift", or "trend" for regression model
# type: “无”，“漂移”，或“趋势”回归模型
# lags: number of lags or max number of lags if using selectlags
# lags: 滞后量，或在设置selectlags参数后的最大滞后量
# selectlags: "fixed", or either "AIC" or "BIC" for auto-selection
# selectlags: "fixed", 或者用于自动选择的"AIC"或"BIC"

# Example:
library(FinTS)
data(d.sp9003lev)
sp500.level <- window(d.sp9003lev,start="1995-01-01")
(ht <- ur.df(sp500.level,type="trend",lags=20,selectlags="BIC"))
####################################### 
# Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
####################################### 
# The value of the test statistic is: -1.4807 1.631 1.8925 
# 检验统计量的值分别为：-1.4807 1.631 1.8925

attributes(ht)$teststat # tau3 = -1.480685
#                tau3 phi2 phi3
# statistic -1.480685 1.631008 1.892498

attributes(ht)$cval # > tau3 5pct = -3.41
#       1pct 5pct 10pct
# tau3 -3.96 -3.41 -3.12
# phi2 6.09 4.68 4.03
# phi3 8.27 6.25 5.34

# Conclusion: We cannot reject the null-hypothesis of a unit root
# (tau3 is less negative than its critical value of -3.41)
# 结论：（tau3没有临界值-3.41显著）

8 Phillips & Ouliaris Test

Null hypothesis: No cointegration. 没有协整关系

install.packages("urca")
library(urca)
args(ca.po)

# function (z, demean = c("none", "constant", "trend"), lag = c("short", 
#     "long"), type = c("Pu", "Pz"), tol = NULL) 
# NULL
# x: matrix or multivariate time series to be tested
# x: 用于测试的矩阵或多维时间序列
# demean: "none", "constant", or "trend"
# lag: "short" or "long" for the length of lags
# lag: 滞后长度，"short"或者"long"
# type: "Pu" or "Pz"

install.packages("tseries")
library(tseries)
args(po.test)
# function (x, demean = TRUE, lshort = TRUE) 
# NULL
# x: matrix or multivariate time series to be tested
# demean: should an intercept be included in the cointegration
# demean: 协整是否包含截距
# regression: (T/F)
# lshort: T = short number of lags, F = long number of lags
# lshort: T = 短滞后阶数, F = 长滞后阶数

# Example:
data(ecb) # Macroeconomic data of the Euro Zone
# 欧元区宏观经济数据
m3.real <- ecb[,"m3"]/ecb[,"gdp.defl"]
gdp.real <- ecb[,"gdp.nom"]/ecb[,"gdp.defl"]
rl <- ecb[,"rl"]
ecb.data <- cbind(m3.real, gdp.real, rl)

(m3d.po <- ca.po(ecb.data, type="Pz"))
################################# 
# Phillips and Ouliaris Unit Root / Cointegration Test # 
################################# 
# The value of the test statistic is: 18.4658 
slotNames(m3d.po)
# [1] "z" "type" "model" "lag" "cval" 
# [6] "res" "teststat" "testreg" "test.name"
m3d.po@teststat
# [1] 18.46584
m3d.po@cval
#                   10pct 5pct 1pct
# critical values 62.1436 71.2751 89.6679
# Conclusion: We should not reject the null hypothesis.
# (not as extreme as its critical value)
# (不如临界值显著)

(m3d.po <- po.test(ecb.data))
#         Phillips-Ouliaris Cointegration Test
# data: ecb.data
# Phillips-Ouliaris demeaned = -4.4665, Truncation lag
# parameter = 0, p-value = 0.15

names(m3d.po)
# [1] "statistic" "parameter" "p.value" "method" "data.name"

m3d.po$p.value
# [1] 0.15
# Conclusion: We should not reject the null hypothesis.

9 Johansen Procedure Test

Null hypothesis: There are n cointegrated vectors. 有n个协整向量

install.packages("urca")
library(urca)
args(ca.jo)
# function (x, type = c("eigen", "trace"), ecdet = c("none", "const", 
#     "trend"), K = 2, spec = c("longrun", "transitory"), season = NULL, 
#     dumvar = NULL) 
# NULL
# x: data matrix to be investigated for cointegration
# x: 用于检查协整的数据矩阵
# type: "eigen" or "trace"
# ecdet: "none", "const", or "trend"
# K: number of lags
# K: 滞后阶数
# spec: VECM specification, "longrun" or "transitory"
# spec: VECM规范："longrun"或"transitory"

# Example:
data(denmark)
sjd <- denmark[, c("LRM", "LRY", "IBO", "IDE")]
(sjd.vecm <- ca.jo(sjd, ecdet = "const", type="eigen", K=2,
+ spec="longrun",season=4))

################################# 
# Johansen-Procedure Unit Root / Cointegration Test # 
################################# 
# The value of the test statistic is: 2.3522 6.3427 10.362 30.0875 
summary(sjd.vecm)
###################### 
# Johansen-Procedure # 
###################### 
# Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration 
# 测试类型：没有线性趋势并协整恒定的最大特征值统计（λ最大）
# Eigenvalues (lambda):
# 特征值（λ）：
# [1] 4.331654e-01 1.775836e-01 1.127905e-01 4.341130e-02
# [5] -8.947236e-16
# Values of teststatistic and critical values of test:
# 测试检验统计量的值和临界值：
#           test 10pct 5pct 1pct
# r <= 3 | 2.35 7.52 9.24 12.97
# r <= 2 | 6.34 13.75 15.67 20.20
# r <= 1 | 10.36 19.77 22.00 26.81
# r = 0 | 30.09 25.56 28.14 33.24
# Eigenvectors, normalised to first column:
# 归一化的特征向量
# (These are the cointegration relations)
# 没有协整关系
#             LRM.l2 LRY.l2 IBO.l2 IDE.l2 constant
# LRM.l2 1.000000 1.0000000 1.0000000 1.000000 1.0000000
# LRY.l2 -1.032949 -1.3681031 -3.2266580 -1.883625 -0.6336946
# IBO.l2 5.206919 0.2429825 0.5382847 24.399487 1.6965828
# IDE.l2 -4.215879 6.8411103 -5.6473903 -14.298037 -1.8951589
# constant -6.059932 -4.2708474 7.8963696 -2.263224 -8.0330127
# Weights W:
# (This is the loading matrix)
# （这是载荷矩阵）
#            LRM.l2 LRY.l2 IBO.l2 IDE.l2
# LRM.d -0.21295494 -0.00481498 0.035011128 2.028908e-03
# LRY.d 0.11502204 0.01975028 0.049938460 1.108654e-03
# IBO.d 0.02317724 -0.01059605 0.003480357 -1.573742e-03
# IDE.d 0.02941109 -0.03022917 -0.002811506 -4.767627e-05
#            constant
# LRM.d -4.183363e-13
# LRY.d 4.618135e-13
# IBO.d 3.673136e-14
# IDE.d -1.115087e-14

slotNames(sjd.vecm)
#  [1] "x" "Z0" "Z1" "ZK" "type" 
#  [6] "model" "ecdet" "lag" "P" "season" 
# [11] "dumvar" "cval" "teststat" "lambda" "Vorg" 
# [16] "V" "W" "PI" "DELTA" "GAMMA" 
# [21] "R0" "RK" "bp" "spec" "call" 
# [26] "test.name"
sjd.vecm@teststat
# [1] 2.352233 6.342730 10.361950 30.087451
sjd.vecm@cval # 10.36 < r<=1 = 22.00 
#          10pct 5pct 1pct
# r <= 3 | 7.52 9.24 12.97
# r <= 2 | 13.75 15.67 20.20
# r <= 1 | 19.77 22.00 26.81
# r = 0 | 25.56 28.14 33.24
# Conclusion: There are 1 cointegrated vector.
# 结论：有1个协整向量。

10 K检验(经验分布的Kolmogorov-Smirnov检验)

R函数:ks.test()

如果P值很小，说明拒绝原假设，表明数据不符合F(n,m)分布。

11 T-test T检验

用于正态总体均值假设检验，单样本，双样本都可以。可以双边检验，也可以单边检验。

对于单样本，与给定的均值做比较： （1）双边检验，也就是 Null hypothesis: 这个样本的均值与给定均值相等：H0： u = u0

Alternative hypothesis: 这个样本的均值与给定均值不相等：H1： u ≠ u0

（2）单边检验，也就是 单边检验I Null hypothesis: 这个样本的均值小于给定均值：H0： u ≤ u0

Alternative hypothesis: 这个样本的均值大于给定均值：H1： u ＞ u0

单边检验II（单边检验I反过来） Null hypothesis: 这个样本的均值大于等于给定均值：H0： u ≥ u0

Alternative hypothesis: 这个样本的均值小于给定均值：H1： u ＜ u0

对于双样本，就是两个样本的均值做比较。单边、双边检验与单样本一样。

结果意义：P值小于显著性水平时拒绝原假设，否则，接受原假设。具体的假设要看所选择的是双边假设还是单边假设（又分小于和大于）

t.test(x, y = NULL,
alternative = c("two.sided", "less", "greater"),
mu = 0, paired = FALSE, var.equal = FALSE,
conf.level = 0.95, ...)

12 正态总体方差检验

用于正态总体方差假设检验，单样本，双样本都可以。可以双边检验，也可以单边检验。与均值假设检验类似，不过检验的对象变成方差了。

结果含义：P值小于显著性水平时拒绝原假设，否则，接受原假设。具体的假设要看所选择的是双边假设还是单边假设（又分小于和大于）

t.test(x, y = NULL,
alternative = c("two.sided", "less", "greater"),
mu = 0, paired = FALSE, var.equal = FALSE,
conf.level = 0.95, ...)

13 二项分布总体假设检验

binom.test(x, n, p = 0.5,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95)

14 Pearson 拟合优度χ2检验

原假设H0：X符合F分布。 p-值小于某个显著性水平，则表示拒绝原假设，否则接受原假设。

chisq.test(x, y = NULL, correct = TRUE,
p = rep(1/length(x), length(x)), rescale.p = FALSE,
simulate.p.value = FALSE, B = 2000)

15 Fisher精确的独立检验：

原假设：X,Y相关

fisher.test(x, y = NULL, workspace = 200000, hybrid = FALSE,
control = list(), or = 1, alternative = "two.sided",
conf.int = TRUE, conf.level = 0.95)

16 McNemar检验：

原假设：两组数据的频数没有区别。

mcnemar.test(x, y = NULL, correct = TRUE)

17 秩相关检验

原假设：x,y相关.

cor.test(x, y,
alternative = c("two.sided", "less", "greater"),
method = "spearman", conf.level = 0.95, ...)

18 Wilcoxon秩检验

原假设：中位数大于，小于，不等于mu.

wilcox.test(x, y = NULL,
alternative = c("two.sided", "less", "greater"),
mu = 0, paired = FALSE, exact = NULL, correct = TRUE,
conf.int = FALSE, conf.level = 0.95, ...)