# Integrated Modified OLS

### Description

Computes the Vogelsang and Wagner (2014) Integrated Modified OLS estimator.

### Usage

1 2 |

### Arguments

`x` |
[ |

`y` |
[ |

`deter` |
[ |

`selector` |
[ |

`t.test` |
[ |

`kernel` |
[ |

`bandwidth` |
[ |

`check` |
[ |

`...` |
Arguments passed to |

### Details

The equation for which the IM-OLS estimator is calculated (type 1):

*
S[y] = δ * S[D] + β * S[x] + γ * x + u*

where *S[y]*, *S[x]* and *S[D]* are the cumulated
sums of *y*, *x* and *D* (with *D* as the deterministics
matrix).
Then *θ = (δ', β', γ')'* is the full parameter vector.

The equation for which the IM-OLS estimator is calculated (type 2):

*S[y] = δ * S[D] + β * S[x] + γ * x +
λ * Z + u*

where *S[y]*, *S[x]* and *S[D]* are the cumulated
sums of *y*, *x* and *D* (with *D* as the deterministics
matrix) and *Z* as defined in equation (19) in Vogelsang and Wagner
(2015).
Then *θ = (δ', β', γ', λ')'* is the full
parameter vector.

### Value

[`cointReg`

]. List with components:

`delta`

[`numeric`

]-
coefficients of the deterministics (cumulative sum

*S_{deter}*) `beta`

[`numeric`

]-
coefficients of the regressors (cumulative sum

*S_{x}*) `gamma`

[`numeric`

]-
coefficients of the regressors (original regressors

*x*) `theta`

[`numeric`

]-
combined coefficients of

`beta`

,`delta`

`sd.theta`

[`numeric`

]-
standard errors for the

`theta`

coefficients `t.theta`

[`numeric`

]-
t-values for the

`theta`

coefficients `p.theta`

[`numeric`

]-
p-values for the

`theta`

coefficients `theta.all`

[`numeric`

]-
combined coefficients of

`beta`

,`delta`

,`gamma`

`residuals`

[`numeric`

]-
IM-OLS residuals. Attention: These are the first differences of

*S_u*– the original residuals are stored in`u.plus`

. `u.plus`

[`numeric`

]-
IM-OLS residuals, not differenced. See

`residuals`

above. `omega.u.v`

[`numeric`

]-
conditional long-run variance based on OLS residuals, via

`cointRegFM`

(in case of argument`t.test`

is`TRUE`

) or`NULL`

`varmat`

[`matrix`

]-
variance-covariance matrix

`Omega`

[`matrix`

]-
`NULL`

(no long-run variance matrix for this regression type) `bandwidth`

[`list`

]-
`number`

and`name`

of bandwidth if`t.test = TRUE`

`kernel`

[`character`

]-
abbr. name of kernel type if

`t.test = TRUE`

`delta2`

[`numeric`

]-
coefficients of the deterministics (cumulative sum

*S_{deter}*) for regression type 2 `beta2`

[`numeric`

]-
coefficients of the regressors (cumulative sum

*S_{x}*) for regression type 2 `gamma2`

[`numeric`

]-
coefficients of the regressors (original regressors

*x*) for regression type 2 `lambda2`

[`numeric`

]-
coefficients of the Z regressors for regression type 2

`theta2`

[`numeric`

]-
combined coefficients of

`beta2`

,`delta2`

,`gamma2`

and`lambda2`

for regression type 2 `u.plus2`

[`numeric`

]-
IM-OLS residuals for regression type 2

### References

Vogelsang, T.J. and M. Wagner (2014): "Integrated Modified OLS Estimation and Fixed-b Inference for Cointegrating Regressions,"

*Journal of Econometrics*, 148, 741–760, DOI:10.1016/j.jeconom.2013.10.015.

### See Also

Other cointReg: `cointRegD`

,
`cointRegFM`

, `cointReg`

,
`plot.cointReg`

, `print.cointReg`

### Examples

1 2 3 4 5 6 7 8 9 10 | ```
set.seed(1909)
x1 = cumsum(rnorm(100, mean = 0.05, sd = 0.1))
x2 = cumsum(rnorm(100, sd = 0.1)) + 1
x3 = cumsum(rnorm(100, sd = 0.2)) + 2
x = cbind(x1, x2, x3)
y = x1 + x2 + x3 + rnorm(100, sd = 0.2) + 1
deter = cbind(level = 1, trend = 1:100)
test = cointRegIM(x, y, deter, selector = c(1, 2), t.test = TRUE,
kernel = "ba", bandwidth = "and")
print(test)
``` |