目录

原文博客地址：https://blog.csdn.net/s1164548515/article/details/101011785

示例：

R代码实现

加载包

生成符合条件的随机数

权重分配：Exponential Almon polynomial 约束一致系数

低频序列模拟 (e.g. 年度)

MIDAS 回归示例 月度、季度数据转化为同频

基于最小二乘的线性模型

基于无约束的混频回归

基于midas_r的非线性估计

收敛性检验

其它加权形式

约束的充分性检验

最优模型选取

手动选取模型

示例：

根据上述函数及数据要求，利用R代码实现混频回归示例。具体代码如下：

R代码实现

加载包

library(midasr)

生成符合条件的随机数

  1. set.seed(1001) # 设置随机数种子
  2. n <- 250#低频数据样本容量
  3. trend <- c(1:n)#线性趋势
  4. x <- rnorm(4 * n)#高频解释变量，eg:季度数据
  5. z <- rnorm(12 * n)#高频解释变量，eg:月度数据

权重分配：Exponential Almon polynomial 约束一致系数

常见加权函数形式：

  1. #权重分配：指数阿尔蒙滞后多项式nealmon
  2. fn_x <- nealmon(p = c(1, -0.5), d = 8)
  3. fn_z <- nealmon(p = c(2, 0.5, -0.1), d = 17)

低频序列模拟 (e.g. 年度)

y <- 2 + 0.1 * trend + mls(x, 0:7, 4) %*% fn_x + mls(z, 0:16, 12) %*% fn_z + rnorm(n)

绘制系数散点图

  1. plot(fn_z, col = "red")
  2. points(fn_x,col = "blue")

MIDAS 回归示例 月度、季度数据转化为同频

基于最小二乘的线性模型

  1. eq_u1 <- lm(y ~ trend + mls(x, k = 0:7, m = 4) + mls(z, k = 0:16, m = 12))
  2. summary(eq_u1)

eq_u1结果：

  1. Call:
  2. lm(formula = y ~ trend + mls(x, k = 0:7, m = 4) + mls(z, k = 0:16,
  3. m = 12))
  4. Residuals:
  5. Min 1Q Median 3Q Max
  6. -2.2651 -0.6489 0.1073 0.6780 2.7707
  7. Coefficients:
  8. Estimate Std. Error t value Pr(>|t|)
  9. (Intercept) 1.9694327 0.1261210 15.615 < 2e-16 ***
  10. trend 0.1000072 0.0008769 114.047 < 2e-16 ***
  11. mls(x, k = 0:7, m = 4)X.0/m 0.5268124 0.0643322 8.189 2.07e-14 ***
  12. mls(x, k = 0:7, m = 4)X.1/m 0.3782006 0.0641497 5.896 1.38e-08 ***
  13. mls(x, k = 0:7, m = 4)X.2/m 0.1879689 0.0680465 2.762 0.006219 **
  14. mls(x, k = 0:7, m = 4)X.3/m -0.0052409 0.0658730 -0.080 0.936658
  15. mls(x, k = 0:7, m = 4)X.4/m 0.1504419 0.0627623 2.397 0.017358 *
  16. mls(x, k = 0:7, m = 4)X.5/m 0.0104345 0.0655386 0.159 0.873647
  17. mls(x, k = 0:7, m = 4)X.6/m 0.0698753 0.0692803 1.009 0.314270
  18. mls(x, k = 0:7, m = 4)X.7/m 0.1463317 0.0650285 2.250 0.025412 *
  19. mls(z, k = 0:16, m = 12)X.0/m 0.3671055 0.0618960 5.931 1.14e-08 ***
  20. mls(z, k = 0:16, m = 12)X.1/m 0.3502401 0.0599615 5.841 1.83e-08 ***
  21. mls(z, k = 0:16, m = 12)X.2/m 0.4514656 0.0659546 6.845 7.37e-11 ***
  22. mls(z, k = 0:16, m = 12)X.3/m 0.3733747 0.0577702 6.463 6.42e-10 ***
  23. mls(z, k = 0:16, m = 12)X.4/m 0.3609667 0.0700954 5.150 5.75e-07 ***
  24. mls(z, k = 0:16, m = 12)X.5/m 0.2155748 0.0632036 3.411 0.000769 ***
  25. mls(z, k = 0:16, m = 12)X.6/m 0.0648163 0.0630194 1.029 0.304828
  26. mls(z, k = 0:16, m = 12)X.7/m 0.0665581 0.0673284 0.989 0.323955
  27. mls(z, k = 0:16, m = 12)X.8/m -0.0014853 0.0624569 -0.024 0.981048
  28. mls(z, k = 0:16, m = 12)X.9/m 0.0466486 0.0675116 0.691 0.490305
  29. mls(z, k = 0:16, m = 12)X.10/m 0.0384882 0.0664136 0.580 0.562824
  30. mls(z, k = 0:16, m = 12)X.11/m -0.0077722 0.0591409 -0.131 0.895564
  31. mls(z, k = 0:16, m = 12)X.12/m -0.0283221 0.0620632 -0.456 0.648589
  32. mls(z, k = 0:16, m = 12)X.13/m -0.0375062 0.0608348 -0.617 0.538179
  33. mls(z, k = 0:16, m = 12)X.14/m 0.0297271 0.0652273 0.456 0.649018
  34. mls(z, k = 0:16, m = 12)X.15/m 0.0184075 0.0588059 0.313 0.754558
  35. mls(z, k = 0:16, m = 12)X.16/m -0.0546460 0.0677214 -0.807 0.420574
  36. ---
  37. Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  38. Residual standard error: 0.9383 on 222 degrees of freedom
  39. (1 observation deleted due to missingness)
  40. Multiple R-squared: 0.9855, Adjusted R-squared: 0.9838
  41. F-statistic: 579.3 on 26 and 222 DF, p-value: < 2.2e-16

基于无约束的混频回归

  1. eq_u2 <- midas_u(y ~ trend + mls(x, 0:7, 4) + mls(z, 0:16, 12))
  2. summary(eq_u2)

eq_u2结果：

  1. Call:
  2. lm(formula = y ~ trend + mls(x, 0:7, 4) + mls(z, 0:16, 12), data = ee)
  3. Residuals:
  4. Min 1Q Median 3Q Max
  5. -2.2651 -0.6489 0.1073 0.6780 2.7707
  6. Coefficients:
  7. Estimate Std. Error t value Pr(>|t|)
  8. (Intercept) 1.9694327 0.1261210 15.615 < 2e-16 ***
  9. trend 0.1000072 0.0008769 114.047 < 2e-16 ***
  10. mls(x, 0:7, 4)X.0/m 0.5268124 0.0643322 8.189 2.07e-14 ***
  11. mls(x, 0:7, 4)X.1/m 0.3782006 0.0641497 5.896 1.38e-08 ***
  12. mls(x, 0:7, 4)X.2/m 0.1879689 0.0680465 2.762 0.006219 **
  13. mls(x, 0:7, 4)X.3/m -0.0052409 0.0658730 -0.080 0.936658
  14. mls(x, 0:7, 4)X.4/m 0.1504419 0.0627623 2.397 0.017358 *
  15. mls(x, 0:7, 4)X.5/m 0.0104345 0.0655386 0.159 0.873647
  16. mls(x, 0:7, 4)X.6/m 0.0698753 0.0692803 1.009 0.314270
  17. mls(x, 0:7, 4)X.7/m 0.1463317 0.0650285 2.250 0.025412 *
  18. mls(z, 0:16, 12)X.0/m 0.3671055 0.0618960 5.931 1.14e-08 ***
  19. mls(z, 0:16, 12)X.1/m 0.3502401 0.0599615 5.841 1.83e-08 ***
  20. mls(z, 0:16, 12)X.2/m 0.4514656 0.0659546 6.845 7.37e-11 ***
  21. mls(z, 0:16, 12)X.3/m 0.3733747 0.0577702 6.463 6.42e-10 ***
  22. mls(z, 0:16, 12)X.4/m 0.3609667 0.0700954 5.150 5.75e-07 ***
  23. mls(z, 0:16, 12)X.5/m 0.2155748 0.0632036 3.411 0.000769 ***
  24. mls(z, 0:16, 12)X.6/m 0.0648163 0.0630194 1.029 0.304828
  25. mls(z, 0:16, 12)X.7/m 0.0665581 0.0673284 0.989 0.323955
  26. mls(z, 0:16, 12)X.8/m -0.0014853 0.0624569 -0.024 0.981048
  27. mls(z, 0:16, 12)X.9/m 0.0466486 0.0675116 0.691 0.490305
  28. mls(z, 0:16, 12)X.10/m 0.0384882 0.0664136 0.580 0.562824
  29. mls(z, 0:16, 12)X.11/m -0.0077722 0.0591409 -0.131 0.895564
  30. mls(z, 0:16, 12)X.12/m -0.0283221 0.0620632 -0.456 0.648589
  31. mls(z, 0:16, 12)X.13/m -0.0375062 0.0608348 -0.617 0.538179
  32. mls(z, 0:16, 12)X.14/m 0.0297271 0.0652273 0.456 0.649018
  33. mls(z, 0:16, 12)X.15/m 0.0184075 0.0588059 0.313 0.754558
  34. mls(z, 0:16, 12)X.16/m -0.0546460 0.0677214 -0.807 0.420574
  35. ---
  36. Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  37. Residual standard error: 0.9383 on 222 degrees of freedom
  38. (1 observation deleted due to missingness)
  39. Multiple R-squared: 0.9855, Adjusted R-squared: 0.9838
  40. F-statistic: 579.3 on 26 and 222 DF, p-value: < 2.2e-16

同频时基于最小二乘和无约束混频回归的eq_u1与eq_u2结果一致！

基于midas_r的非线性估计

  1. eq_r <- midas_r(y ~ trend + mls(x, 0:7, 4, nealmon) + mls(z, 0:16, 12, nealmon), start = list(x = c(1,-0.5), z = c(2, 0.5, -0.1)))
  2. summary(eq_r)

收敛性检验

通过计算梯度和黑森矩阵检验是否满足收敛的充分必要条件，计算梯度的欧氏范数和hessian的特征值。然后判断梯度范数是否接近零，特征值是否为正

deriv_tests(eq_r, tol = 1e-06)

结果：

  1. $`first`
  2. [1] FALSE
  3. $second
  4. [1] TRUE
  5. $gradient
  6. [1] 0.005542527 0.173490055 -0.005754945 0.028655981 -0.030150807 0.019224017 0.053829528
  7. $eigenval
  8. [1] 1.047988e+07 5.891393e+04 3.664513e+02 1.221231e+02 8.116275e+01 5.148844e+01 4.598507e+01

系数输出

coef(eq_r)#返回t检验显著的系数

结果：

  1. (Intercept) trend x1 x2 z1 z2 z3
  2. 1.98820762 0.09988275 1.35336157 -0.50747744 2.26312231 0.40927986 -0.07293300 
coef(eq_r, midas = TRUE)#返回所有系数

结果：

  1. (Intercept) trend x1 x2 x3 x4 x5 x6 x7
  2. 1.988208e+00 9.988275e-02 5.480768e-01 3.299490e-01 1.986333e-01 1.195797e-01 7.198845e-02 4.333793e-02 2.608997e-02
  3. x8 z1 z2 z3 z4 z5 z6 z7 z8
  4. 1.570648e-02 3.346806e-01 4.049071e-01 4.233810e-01 3.826120e-01 2.988388e-01 2.017282e-01 1.176921e-01 5.934435e-02
  5. z9 z10 z11 z12 z13 z14 z15 z16 z17
  6. 2.586203e-02 9.740853e-03 3.170901e-03 8.921121e-04 2.169239e-04 4.558760e-05 8.280130e-06 1.299807e-06 1.763484e-07

eq_r的基础上使用函数amweights来形成几个标准的周期函数约束

  1. amweights(p = c(1, -0.5), d = 8, m = 4, weight = nealmon, type = "C")
  2. nealmon(p = c(1, -0.5), d = 4)
  3. eq_r1 <- midas_r(y ~ trend + mls(x, 0:7, 4, amweights, nealmon, "C") + mls(z, 0:16, 12, nealmon), start = list(x = c(1, -0.5), z = c(2, 0.5, -0.1)))
  4. summary(eq_r1)

其它加权形式

  1. fn <- gompertzp#概率密度函数的权重设定
  2. eq_r2 <- midas_r(y ~ trend + mls(x, 0:7, 4, nealmon) + mls(z, 0:16, 12, fn), start = list(x = c(1,-0.5), z = c(1, 0.5, 0.1)))
  3. summary(eq_r2)
  1. eq_r3 <- midas_r(y ~ trend + mls(x, 0:7, 4) + mls(z, 0:16, 12, nealmon), start = list(z = c(1,-0.5)))
  2. summary(eq_r3)
  3. eq_r4 <- midas_r(y ~ trend + mls(y, 1:2, 1) + mls(x, 0:7, 4, nealmon), start = list(x = c(1,-0.5)))
  4. summary(eq_r4)
  5. eq_r5 <- midas_r(y ~ trend + mls(y, 1:2, 1, "*") + mls(x, 0:7, 4, nealmon), start = list(x = c(1,-0.5)))
  6. summary(eq_r5)
  7. eq_r6 <- midas_r(y ~ trend + mls(y, 1:4, 1, nealmon) + mls(x, 0:7, 4, nealmon), start = list(y = c(1, -0.5), x = c(1, -0.5)))
  8. summary(eq_r6)
  9. eq_r7 <- midas_r(y ~ trend + mls(x, 0:7, 4, amweights, nealmon, "B"), start = list(x = c(1, 1, -0.5))) summary(eq_r7)
  10. eq_r8 <- midas_r(y ~ trend + mls(x, 0:7, 4, amweights, nealmon, "C"), start = list(x = c(1, -0.5))) summary(eq_r8)
  11. eq_r9 <- midas_r(y ~ trend + mls(x, 0:7, 4, amweights, nealmon, "A"), start = list(x = c(1, 1, 1, -0.5)))
  12. summary(eq_r9)
  1. fn <- function(p, d) {
  2. p[1] * c(1:d)^p[2]
  3. }
  4. eq_r10 <- midas_r(y ~ trend + mls(x, 0:101, 4, fn), start = list(x = rep(0, 2)))
  5. summary(eq_r10)

约束的充分性检验

只要误差是独立同分布的，就可以使用hAh_test检验，而hAhr_test是HAC-Roust稳健版本检验。只要在残差中没有观察到显著的hac，在小样本中使用hAh_test检验更精确。

  1. eq_r12 <- midas_r(y ~ trend + mls(x, 0:7, 4, nealmon) + mls(z, 0:16, 12, nealmon), start = list(x = c(1, -0.5), z = c(2, 0.5, -0.1)))
  2. summary(eq_r12)
  3. #约束的充分性检验
  4. hAh_test(eq_r12)
  5. hAhr_test(eq_r12)

结果：

  1. hAh restriction test
  2. data:
  3. hAh = 16.552, df = 20, p-value = 0.6818
  4. hAh restriction test (robust version)
  5. data:
  6. hAhr = 14.854, df = 20, p-value = 0.7847

对比：将变量z的函数约束的参数数量由17减少为2个

  1. eq_r13 <- midas_r(y ~ trend + mls(x, 0:7, 4, nealmon) + mls(z, 0:12, 12, nealmon), start = list(x = c(1, -0.5), z = c(2, -0.1)))
  2. hAh_test(eq_r13)
  3. hAhr_test(eq_r13)
  4. summary(eq_r13)

结果：

  1. hAh restriction test
  2. data:
  3. hAh = 36.892, df = 17, p-value = 0.00348
  4. hAh restriction test (robust version)
  5. data:
  6. hAhr = 32.879, df = 17, p-value = 0.01168

拒绝原假设，即约束不充分

最优模型选取

step1:函数expand_weights_lags定义每个模型的潜在模型集

  1. set_x <- expand_weights_lags(weights = c("nealmon", "almonp"), from = 0, to = c(5, 10), m = 1, start = list(nealmon = rep(1, 2), almonp = rep(1, 3)))
  2. set_z <- expand_weights_lags(weights = c("nealmon", "nbeta"), from = 1, to = c(2, 3), m = 1, start = list(nealmon = rep(0,2), nbeta = rep(0.5, 3)))

step2:所有可能模型的估计

函数midas_r_ic_table返回所有模型的汇总表，以及常用信息准则的相应值和参数约束充分性检验（默认是hAh_test检验）的经验大小。给出了导数检验的结果和优化函数的收敛状态。

  1. eqs.ic <- midas_r_ic_table(y ~ trend + mls(x, 0, m = 4) + fmls(z, 0, m = 12), table = list(z = set_z,x = set_x), start = c(`(Intercept)` = 0, trend = 0))
  2. mod <- modsel(eqs.ic, IC = "AIC", type = "restricted")

step3：查看某个候选模型可微调其收敛性，并更新结果：

  1. eqs_ic$candlist[[5]] <- update(eqs_ic$candlist[[5]], Ofunction = "nls")
  2. eqs_ic <- update(eqs_ic)

手动选取模型

step1:训练集（样本内预测）、测试集（样本外预测）分割

datasplit <- split_data(list(y = y, x = x, z = z, trend = trend), insample = 1:200, outsample = 201:250)

step2:分别拟合待选模型

  1. mod1 <- midas_r(y ~ trend + mls(x, 4:14, 4, nealmon) + mls(z, 12:22, 12, nealmon), data = datasplit$indata, start = list(x = c(10, 1, -0.1), z = c(2, -0.1)))
  2. mod2 <- midas_r(y ~ trend + mls(x, 4:20, 4, nealmon) + mls(z, 12:25, 12, nealmon), data = datasplit$indata, start = list(x = c(10, 1, -0.1), z = c(2, -0.1)))

step3:计算

avgf <- average_forecast(list(mod1, mod2), data = list(y = y, x = x, z = z, trend = trend), insample = 1:200, outsample = 201:250, type = "fixed", measures = c("MSE", "MAPE", "MASE"), fweights = c("EW", "BICW", "MSFE", "DMSFE"))

样本内、样本外预测精度输出：

avgf$accuracy

结果对比：

  1. $`individual`
  2. Model MSE.out.of.sample MAPE.out.of.sample
  3. 1 y ~ trend + mls(x, 4:14, 4, nealmon) + mls(z, 12:22, 12, nealmon) 1.848474 4.474242
  4. 2 y ~ trend + mls(x, 4:20, 4, nealmon) + mls(z, 12:25, 12, nealmon) 1.763432 4.391594
  5. MASE.out.of.sample MSE.in.sample MAPE.in.sample MASE.in.sample
  6. 1 0.8155477 1.768720 14.05880 0.6992288
  7. 2 0.7976764 1.726365 13.56287 0.6897242
  8. $average
  9. Scheme MSE MAPE MASE
  10. 1 EW 1.803640 4.432918 0.8066121
  11. 2 BICW 1.763433 4.391595 0.7976766
  12. 3 MSFE 1.802640 4.431945 0.8064017
  13. 4 DMSFE 1.801787 4.431112 0.8062216

mod1,mod2预测结果输出：

avgf$forecast

结果对比：

  1. [,1] [,2]
  2. [1,] 22.24057 22.38300
  3. [2,] 22.12774 22.13778
  4. [3,] 22.22678 22.21942
  5. [4,] 22.44535 22.61872
  6. [5,] 22.36448 22.46157
  7. [6,] 22.49847 22.66663
  8. [7,] 22.70192 22.70818
  9. [8,] 22.49411 22.53326
  10. [9,] 22.47978 22.54235
  11. [10,] 22.43439 22.46522
  12. [11,] 22.61563 22.57788
  13. [12,] 22.87686 22.88271
  14. [13,] 22.59329 22.64370
  15. [14,] 23.39675 23.47606
  16. [15,] 23.25864 23.27932
  17. [16,] 22.97588 23.09488
  18. [17,] 23.13196 23.21956
  19. [18,] 24.05489 24.04334
  20. [19,] 24.11690 24.16742
  21. [20,] 23.79328 23.91429
  22. [21,] 24.58047 24.58925
  23. [22,] 24.33794 24.45825
  24. [23,] 23.96658 24.03107
  25. [24,] 24.00510 24.12939
  26. [25,] 23.96473 23.99645
  27. [26,] 24.02198 24.04416
  28. [27,] 25.01521 24.91966
  29. [28,] 24.47063 24.59918
  30. [29,] 24.58748 24.51964
  31. [30,] 24.69662 24.72768
  32. [31,] 25.19092 25.22659
  33. [32,] 25.57452 25.72345
  34. [33,] 25.06312 25.17678
  35. [34,] 24.99302 25.09359
  36. [35,] 25.13005 25.16293
  37. [36,] 25.85870 25.85633
  38. [37,] 25.43593 25.46734
  39. [38,] 25.80690 25.88224
  40. [39,] 26.80900 26.82394
  41. [40,] 26.44056 26.63095
  42. [41,] 26.04786 26.17996
  43. [42,] 26.09545 26.29647
  44. [43,] 25.99309 25.98112
  45. [44,] 25.89755 26.03643
  46. [45,] 25.99814 26.18442
  47. [46,] 26.25727 26.38797
  48. [47,] 26.79563 26.92593
  49. [48,] 26.57034 26.68217
  50. [49,] 26.53808 26.51001
  51. [50,] 26.72772 26.76183

组合预测效果输出：

avgf$avgforecast

结果对比：

其中EW:基于特征值加权组合；BICW:BIC信息准则加权组合；MSFE: 预测均方误差加权组合;DMSFE：discounted MSFE

  1. EW BICW MSFE DMSFE
  2. [1,] 22.31178 22.38299 22.31346 22.31489
  3. [2,] 22.13276 22.13778 22.13288 22.13298
  4. [3,] 22.22310 22.21942 22.22301 22.22294
  5. [4,] 22.53204 22.61872 22.53408 22.53582
  6. [5,] 22.41303 22.46157 22.41417 22.41515
  7. [6,] 22.58255 22.66663 22.58453 22.58622
  8. [7,] 22.70505 22.70818 22.70512 22.70519
  9. [8,] 22.51369 22.53326 22.51415 22.51454
  10. [9,] 22.51107 22.54235 22.51180 22.51244
  11. [10,] 22.44980 22.46522 22.45017 22.45048
  12. [11,] 22.59675 22.57788 22.59631 22.59593
  13. [12,] 22.87979 22.88271 22.87986 22.87991
  14. [13,] 22.61849 22.64370 22.61909 22.61960
  15. [14,] 23.43641 23.47606 23.43734 23.43814
  16. [15,] 23.26898 23.27932 23.26922 23.26943
  17. [16,] 23.03538 23.09488 23.03678 23.03798
  18. [17,] 23.17576 23.21956 23.17679 23.17767
  19. [18,] 24.04912 24.04334 24.04898 24.04887
  20. [19,] 24.14216 24.16742 24.14276 24.14326
  21. [20,] 23.85379 23.91429 23.85521 23.85643
  22. [21,] 24.58486 24.58925 24.58497 24.58505
  23. [22,] 24.39809 24.45825 24.39951 24.40072
  24. [23,] 23.99882 24.03107 23.99958 24.00023
  25. [24,] 24.06724 24.12938 24.06870 24.06996
  26. [25,] 23.98059 23.99645 23.98096 23.98128
  27. [26,] 24.03307 24.04416 24.03334 24.03356
  28. [27,] 24.96744 24.91967 24.96631 24.96535
  29. [28,] 24.53491 24.59918 24.53642 24.53771
  30. [29,] 24.55356 24.51964 24.55277 24.55208
  31. [30,] 24.71215 24.72768 24.71252 24.71283
  32. [31,] 25.20876 25.22659 25.20918 25.20954
  33. [32,] 25.64898 25.72345 25.65074 25.65224
  34. [33,] 25.11995 25.17678 25.12129 25.12244
  35. [34,] 25.04331 25.09359 25.04449 25.04550
  36. [35,] 25.14649 25.16293 25.14688 25.14721
  37. [36,] 25.85751 25.85633 25.85749 25.85746
  38. [37,] 25.45163 25.46734 25.45200 25.45232
  39. [38,] 25.84457 25.88224 25.84546 25.84622
  40. [39,] 26.81647 26.82394 26.81665 26.81680
  41. [40,] 26.53575 26.63095 26.53799 26.53991
  42. [41,] 26.11391 26.17996 26.11546 26.11679
  43. [42,] 26.19596 26.29647 26.19833 26.20035
  44. [43,] 25.98711 25.98112 25.98696 25.98684
  45. [44,] 25.96699 26.03642 25.96862 25.97002
  46. [45,] 26.09128 26.18442 26.09348 26.09535
  47. [46,] 26.32262 26.38797 26.32416 26.32548
  48. [47,] 26.86078 26.92593 26.86232 26.86363
  49. [48,] 26.62626 26.68217 26.62757 26.62870
  50. [49,] 26.52404 26.51001 26.52371 26.52343
  51. [50,] 26.74478 26.76183 26.74518 26.74552

参考文献：《Mixed Frequency Data Sampling Regression Models: The R Package midasr》

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